Theorem elrgspnsubrunlem2 33857

Description: Lemma for elrgspnsubrun 33858, second direction. (Contributed by Thierry Arnoux, 13-Oct-2025.)

Hypotheses

Ref	Expression
elrgspnsubrun.b	⊢ 𝐵 = (Base‘𝑅)
elrgspnsubrun.t	⊢ · = (.r‘𝑅)
elrgspnsubrun.z	⊢ 0 = (0g‘𝑅)
elrgspnsubrun.n	⊢ 𝑁 = (RingSpan‘𝑅)
elrgspnsubrun.r	⊢ (𝜑 → 𝑅 ∈ CRing)
elrgspnsubrun.e	⊢ (𝜑 → 𝐸 ∈ (SubRing‘𝑅))
elrgspnsubrun.f	⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝑅))
elrgspnsubrunlem2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
elrgspnsubrunlem2.1	⊢ (𝜑 → 𝐺:Word (𝐸 ∪ 𝐹)⟶ℤ)
elrgspnsubrunlem2.2	⊢ (𝜑 → 𝐺 finSupp 0)
elrgspnsubrunlem2.3	⊢ (𝜑 → 𝑋 = (𝑅 Σg (𝑤 ∈ Word (𝐸 ∪ 𝐹) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))

Assertion

Ref	Expression
elrgspnsubrunlem2	⊢ (𝜑 → ∃𝑝 ∈ (𝐸 ↑m 𝐹)(𝑝 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑝‘𝑓) · 𝑓)))))

Distinct variable groups:   0 ,𝑓,𝑝,𝑤   · ,𝑓,𝑝,𝑤   𝐵,𝑓,𝑤   𝑓,𝐸,𝑝,𝑤   𝑓,𝐹,𝑝,𝑤   𝑓,𝐺,𝑝,𝑤   𝑅,𝑓,𝑝,𝑤   𝑋,𝑝   𝜑,𝑓,𝑝,𝑤

Allowed substitution hints:   𝐵(𝑝)   𝑁(𝑤, 𝑓, 𝑝)   𝑋(𝑤, 𝑓)

Proof of Theorem elrgspnsubrunlem2

Dummy variables 𝑞 𝑣 𝑦 𝑎 𝑒 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrgspnsubrun.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (SubRing‘𝑅))
2	1	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → 𝐸 ∈ (SubRing‘𝑅))
3		elrgspnsubrun.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝑅))
4	3	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → 𝐹 ∈ (SubRing‘𝑅))
5		elrgspnsubrun.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
6		elrgspnsubrun.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
7	6	crngringd 20480	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
8	7	ringabld 20519	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Abel)
9	8	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → 𝑅 ∈ Abel)
10		vex 3454	. . . . . . . . 9 ⊢ 𝑞 ∈ V
11	10	cnvex 7939	. . . . . . . 8 ⊢ ◡𝑞 ∈ V
12	11	imaex 7928	. . . . . . 7 ⊢ (◡𝑞 “ (𝐸 × {𝑓})) ∈ V
13	12	a1i 11	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (◡𝑞 “ (𝐸 × {𝑓})) ∈ V)
14		subrgsubg 20836	. . . . . . . 8 ⊢ (𝐸 ∈ (SubRing‘𝑅) → 𝐸 ∈ (SubGrp‘𝑅))
15	1, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (SubGrp‘𝑅))
16	15	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → 𝐸 ∈ (SubGrp‘𝑅))
17		elrgspnsubrun.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
18		eqid 2760	. . . . . . . 8 ⊢ (.g‘𝑅) = (.g‘𝑅)
19	6	crnggrpd 20481	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
20	19	ad4antr 745	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑅 ∈ Grp)
21	1, 3	xpexd 7767	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 × 𝐹) ∈ V)
22	1, 3	unexd 7770	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ V)
23		wrdexg 14674	. . . . . . . . . . . . . . 15 ⊢ ((𝐸 ∪ 𝐹) ∈ V → Word (𝐸 ∪ 𝐹) ∈ V)
24	22, 23	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Word (𝐸 ∪ 𝐹) ∈ V)
25	21, 24	elmapd 8865	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹)) ↔ 𝑞:Word (𝐸 ∪ 𝐹)⟶(𝐸 × 𝐹)))
26	25	biimpa 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → 𝑞:Word (𝐸 ∪ 𝐹)⟶(𝐸 × 𝐹))
27	26	ffund 6717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → Fun 𝑞)
28	27	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → Fun 𝑞)
29		fvimacnvi 7054	. . . . . . . . . 10 ⊢ ((Fun 𝑞 ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝑞‘𝑣) ∈ (𝐸 × {𝑓}))
30	28, 29	sylancom 600	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝑞‘𝑣) ∈ (𝐸 × {𝑓}))
31		xp1st 8035	. . . . . . . . 9 ⊢ ((𝑞‘𝑣) ∈ (𝐸 × {𝑓}) → (1st ‘(𝑞‘𝑣)) ∈ 𝐸)
32	30, 31	syl 18	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (1st ‘(𝑞‘𝑣)) ∈ 𝐸)
33	16	adantr 486	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝐸 ∈ (SubGrp‘𝑅))
34		elrgspnsubrunlem2.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:Word (𝐸 ∪ 𝐹)⟶ℤ)
35	34	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝐺:Word (𝐸 ∪ 𝐹)⟶ℤ)
36		cnvimass 6200	. . . . . . . . . . 11 ⊢ (◡𝑞 “ (𝐸 × {𝑓})) ⊆ dom 𝑞
37	26	fdmd 6723	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → dom 𝑞 = Word (𝐸 ∪ 𝐹))
38	37	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → dom 𝑞 = Word (𝐸 ∪ 𝐹))
39	36, 38	sseqtrid 3951	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ Word (𝐸 ∪ 𝐹))
40	39	sselda 3909	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑣 ∈ Word (𝐸 ∪ 𝐹))
41	35, 40	ffvelcdmd 7088	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝐺‘𝑣) ∈ ℤ)
42	17, 18, 20, 32, 33, 41	subgmulgcld 33652	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) ∈ 𝐸)
43	42	fmpttd 7118	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))):(◡𝑞 “ (𝐸 × {𝑓}))⟶𝐸)
44	34	feqmptd 6956	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑣 ∈ Word (𝐸 ∪ 𝐹) ↦ (𝐺‘𝑣)))
45		elrgspnsubrunlem2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 finSupp 0)
46	44, 45	eqbrtrrd 5128	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ Word (𝐸 ∪ 𝐹) ↦ (𝐺‘𝑣)) finSupp 0)
47	46	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ Word (𝐸 ∪ 𝐹) ↦ (𝐺‘𝑣)) finSupp 0)
48		0zd 12710	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → 0 ∈ ℤ)
49	47, 39, 48	fmptssfisupp 9391	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ (𝐺‘𝑣)) finSupp 0)
50	17	subrgss 20831	. . . . . . . . . . 11 ⊢ (𝐸 ∈ (SubRing‘𝑅) → 𝐸 ⊆ 𝐵)
51	1, 50	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ⊆ 𝐵)
52	51	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → 𝐸 ⊆ 𝐵)
53	52	sselda 3909	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑦 ∈ 𝐸) → 𝑦 ∈ 𝐵)
54	17, 5, 18	mulg0 19291	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (0(.g‘𝑅)𝑦) = 0 )
55	53, 54	syl 18	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑦 ∈ 𝐸) → (0(.g‘𝑅)𝑦) = 0 )
56	5	fvexi 6902	. . . . . . . 8 ⊢ 0 ∈ V
57	56	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → 0 ∈ V)
58	49, 55, 41, 32, 57	fsuppssov1 9381	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))) finSupp 0 )
59	5, 9, 13, 16, 43, 58	gsumsubgcl 20141	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) ∈ 𝐸)
60	59	fmpttd 7118	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))):𝐹⟶𝐸)
61	2, 4, 60	elmapdd 8866	. . 3 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))) ∈ (𝐸 ↑m 𝐹))
62		breq1 5105	. . . . 5 ⊢ (𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))) → (𝑝 finSupp 0 ↔ (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))) finSupp 0 ))
63	62	adantl 487	. . . 4 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))) → (𝑝 finSupp 0 ↔ (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))) finSupp 0 ))
64		nfv 1946	. . . . . . . 8 ⊢ Ⅎ𝑓((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤))))
65		nfmpt1 5203	. . . . . . . . 9 ⊢ Ⅎ𝑓(𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))
66	65	nfeq2 2939	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))
67	64, 66	nfan 1931	. . . . . . 7 ⊢ Ⅎ𝑓(((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))))
68		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))) → 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))))
69		ovexd 7457	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))) ∧ 𝑓 ∈ 𝐹) → (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) ∈ V)
70	68, 69	fvmpt2d 7010	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))) ∧ 𝑓 ∈ 𝐹) → (𝑝‘𝑓) = (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))
71	70	oveq1d 7437	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))) ∧ 𝑓 ∈ 𝐹) → ((𝑝‘𝑓) · 𝑓) = ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓))
72	67, 71	mpteq2da 5196	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))) → (𝑓 ∈ 𝐹 ↦ ((𝑝‘𝑓) · 𝑓)) = (𝑓 ∈ 𝐹 ↦ ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓)))
73	72	oveq2d 7438	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))) → (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑝‘𝑓) · 𝑓))) = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓))))
74	73	eqeq2d 2771	. . . 4 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))) → (𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑝‘𝑓) · 𝑓))) ↔ 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓)))))
75	63, 74	anbi12d 644	. . 3 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑝 = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))))) → ((𝑝 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑝‘𝑓) · 𝑓)))) ↔ ((𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))) finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓))))))
76	56	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → 0 ∈ V)
77	60	ffund 6717	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → Fun (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))))
78	27	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → Fun 𝑞)
79	45	fsuppimpd 9366	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 supp 0) ∈ Fin)
80	79	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → (𝐺 supp 0) ∈ Fin)
81		imafi 9312	. . . . . . . 8 ⊢ ((Fun 𝑞 ∧ (𝐺 supp 0) ∈ Fin) → (𝑞 “ (𝐺 supp 0)) ∈ Fin)
82	78, 80, 81	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → (𝑞 “ (𝐺 supp 0)) ∈ Fin)
83		rnfi 9334	. . . . . . 7 ⊢ ((𝑞 “ (𝐺 supp 0)) ∈ Fin → ran (𝑞 “ (𝐺 supp 0)) ∈ Fin)
84	82, 83	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → ran (𝑞 “ (𝐺 supp 0)) ∈ Fin)
85	34	ffnd 6713	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 Fn Word (𝐸 ∪ 𝐹))
86	85	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝐺 Fn Word (𝐸 ∪ 𝐹))
87	24	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → Word (𝐸 ∪ 𝐹) ∈ V)
88		0zd 12710	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 0 ∈ ℤ)
89		snssi 4742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0))) → {𝑓} ⊆ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0))))
90	89	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → {𝑓} ⊆ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0))))
91		xpss2 5670	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑓} ⊆ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0))) → (𝐸 × {𝑓}) ⊆ (𝐸 × (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))))
92		ssun2 4176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸 × (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ⊆ (((𝐸 ∖ dom (𝑞 “ (𝐺 supp 0))) × 𝐹) ∪ (𝐸 × (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))))
93		difxp 6156	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸 × 𝐹) ∖ (dom (𝑞 “ (𝐺 supp 0)) × ran (𝑞 “ (𝐺 supp 0)))) = (((𝐸 ∖ dom (𝑞 “ (𝐺 supp 0))) × 𝐹) ∪ (𝐸 × (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))))
94	92, 93	sseqtrri 3958	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 × (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ⊆ ((𝐸 × 𝐹) ∖ (dom (𝑞 “ (𝐺 supp 0)) × ran (𝑞 “ (𝐺 supp 0))))
95	91, 94	sstrdi 3921	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑓} ⊆ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0))) → (𝐸 × {𝑓}) ⊆ ((𝐸 × 𝐹) ∖ (dom (𝑞 “ (𝐺 supp 0)) × ran (𝑞 “ (𝐺 supp 0)))))
96	90, 95	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝐸 × {𝑓}) ⊆ ((𝐸 × 𝐹) ∖ (dom (𝑞 “ (𝐺 supp 0)) × ran (𝑞 “ (𝐺 supp 0)))))
97		imassrn 6199	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 “ (𝐺 supp 0)) ⊆ ran 𝑞
98	26	frnd 6721	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → ran 𝑞 ⊆ (𝐸 × 𝐹))
99	98	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → ran 𝑞 ⊆ (𝐸 × 𝐹))
100	97, 99	sstrid 3920	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝑞 “ (𝐺 supp 0)) ⊆ (𝐸 × 𝐹))
101		relxp 5668	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Rel (𝐸 × 𝐹)
102		relss 5758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑞 “ (𝐺 supp 0)) ⊆ (𝐸 × 𝐹) → (Rel (𝐸 × 𝐹) → Rel (𝑞 “ (𝐺 supp 0))))
103	101, 102	mpi 21	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑞 “ (𝐺 supp 0)) ⊆ (𝐸 × 𝐹) → Rel (𝑞 “ (𝐺 supp 0)))
104		relssdmrn 6273	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel (𝑞 “ (𝐺 supp 0)) → (𝑞 “ (𝐺 supp 0)) ⊆ (dom (𝑞 “ (𝐺 supp 0)) × ran (𝑞 “ (𝐺 supp 0))))
105	100, 103, 104	3syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝑞 “ (𝐺 supp 0)) ⊆ (dom (𝑞 “ (𝐺 supp 0)) × ran (𝑞 “ (𝐺 supp 0))))
106	105	sscond 4081	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → ((𝐸 × 𝐹) ∖ (dom (𝑞 “ (𝐺 supp 0)) × ran (𝑞 “ (𝐺 supp 0)))) ⊆ ((𝐸 × 𝐹) ∖ (𝑞 “ (𝐺 supp 0))))
107	96, 106	sstrd 3919	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝐸 × {𝑓}) ⊆ ((𝐸 × 𝐹) ∖ (𝑞 “ (𝐺 supp 0))))
108		imass2 6056	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 × {𝑓}) ⊆ ((𝐸 × 𝐹) ∖ (𝑞 “ (𝐺 supp 0))) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ (◡𝑞 “ ((𝐸 × 𝐹) ∖ (𝑞 “ (𝐺 supp 0)))))
109	107, 108	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ (◡𝑞 “ ((𝐸 × 𝐹) ∖ (𝑞 “ (𝐺 supp 0)))))
110	109	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ (◡𝑞 “ ((𝐸 × 𝐹) ∖ (𝑞 “ (𝐺 supp 0)))))
111	78	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → Fun 𝑞)
112		difpreima 7067	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝑞 → (◡𝑞 “ ((𝐸 × 𝐹) ∖ (𝑞 “ (𝐺 supp 0)))) = ((◡𝑞 “ (𝐸 × 𝐹)) ∖ (◡𝑞 “ (𝑞 “ (𝐺 supp 0)))))
113	111, 112	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (◡𝑞 “ ((𝐸 × 𝐹) ∖ (𝑞 “ (𝐺 supp 0)))) = ((◡𝑞 “ (𝐸 × 𝐹)) ∖ (◡𝑞 “ (𝑞 “ (𝐺 supp 0)))))
114		cnvimass 6200	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑞 “ (𝐸 × 𝐹)) ⊆ dom 𝑞
115	37	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → dom 𝑞 = Word (𝐸 ∪ 𝐹))
116	114, 115	sseqtrid 3951	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (◡𝑞 “ (𝐸 × 𝐹)) ⊆ Word (𝐸 ∪ 𝐹))
117		suppssdm 8199	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 supp 0) ⊆ dom 𝐺
118	34	fdmd 6723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom 𝐺 = Word (𝐸 ∪ 𝐹))
119	118	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → dom 𝐺 = Word (𝐸 ∪ 𝐹))
120	117, 119	sseqtrid 3951	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝐺 supp 0) ⊆ Word (𝐸 ∪ 𝐹))
121	120, 115	sseqtrrd 3946	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝐺 supp 0) ⊆ dom 𝑞)
122		sseqin2 4119	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 supp 0) ⊆ dom 𝑞 ↔ (dom 𝑞 ∩ (𝐺 supp 0)) = (𝐺 supp 0))
123	122	biimpi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 supp 0) ⊆ dom 𝑞 → (dom 𝑞 ∩ (𝐺 supp 0)) = (𝐺 supp 0))
124		dminss 6144	. . . . . . . . . . . . . . . . . . 19 ⊢ (dom 𝑞 ∩ (𝐺 supp 0)) ⊆ (◡𝑞 “ (𝑞 “ (𝐺 supp 0)))
125	123, 124	eqsstrrdi 3954	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 supp 0) ⊆ dom 𝑞 → (𝐺 supp 0) ⊆ (◡𝑞 “ (𝑞 “ (𝐺 supp 0))))
126	121, 125	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝐺 supp 0) ⊆ (◡𝑞 “ (𝑞 “ (𝐺 supp 0))))
127	116, 126	ssdif2d 4083	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → ((◡𝑞 “ (𝐸 × 𝐹)) ∖ (◡𝑞 “ (𝑞 “ (𝐺 supp 0)))) ⊆ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0)))
128	113, 127	eqsstrd 3943	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (◡𝑞 “ ((𝐸 × 𝐹) ∖ (𝑞 “ (𝐺 supp 0)))) ⊆ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0)))
129	110, 128	sstrd 3919	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0)))
130	129	sselda 3909	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑣 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0)))
131	86, 87, 88, 130	fvdifsupp 8193	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝐺‘𝑣) = 0)
132	131	oveq1d 7437	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) = (0(.g‘𝑅)(1st ‘(𝑞‘𝑣))))
133	51	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝐸 ⊆ 𝐵)
134	26	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑞:Word (𝐸 ∪ 𝐹)⟶(𝐸 × 𝐹))
135	36, 37	sseqtrid 3951	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ Word (𝐸 ∪ 𝐹))
136	135	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ Word (𝐸 ∪ 𝐹))
137	136	sselda 3909	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑣 ∈ Word (𝐸 ∪ 𝐹))
138	134, 137	ffvelcdmd 7088	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝑞‘𝑣) ∈ (𝐸 × 𝐹))
139		xp1st 8035	. . . . . . . . . . . . . 14 ⊢ ((𝑞‘𝑣) ∈ (𝐸 × 𝐹) → (1st ‘(𝑞‘𝑣)) ∈ 𝐸)
140	138, 139	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (1st ‘(𝑞‘𝑣)) ∈ 𝐸)
141	133, 140	sseldd 3910	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (1st ‘(𝑞‘𝑣)) ∈ 𝐵)
142	17, 5, 18	mulg0 19291	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝑞‘𝑣)) ∈ 𝐵 → (0(.g‘𝑅)(1st ‘(𝑞‘𝑣))) = 0 )
143	141, 142	syl 18	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (0(.g‘𝑅)(1st ‘(𝑞‘𝑣))) = 0 )
144	132, 143	eqtrd 2795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) = 0 )
145	144	mpteq2dva 5197	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))) = (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ 0 ))
146	145	oveq2d 7438	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) = (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ 0 )))
147	19	grpmndd 19164	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Mnd)
148	147	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → 𝑅 ∈ Mnd)
149	12	a1i 11	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (◡𝑞 “ (𝐸 × {𝑓})) ∈ V)
150	5	gsumz 19039	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ (◡𝑞 “ (𝐸 × {𝑓})) ∈ V) → (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ 0 )) = 0 )
151	148, 149, 150	syl2anc 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ 0 )) = 0 )
152	146, 151	eqtrd 2795	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ (𝐹 ∖ ran (𝑞 “ (𝐺 supp 0)))) → (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) = 0 )
153	152, 4	suppss2 8222	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → ((𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))) supp 0 ) ⊆ ran (𝑞 “ (𝐺 supp 0)))
154	84, 153	ssfid 9265	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → ((𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))) supp 0 ) ∈ Fin)
155	61, 76, 77, 154	isfsuppd 9363	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))) finSupp 0 )
156	8	ablcmnd 20009	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CMnd)
157	156	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → 𝑅 ∈ CMnd)
158	24	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → Word (𝐸 ∪ 𝐹) ∈ V)
159	85	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → 𝐺 Fn Word (𝐸 ∪ 𝐹))
160	158	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → Word (𝐸 ∪ 𝐹) ∈ V)
161		0zd 12710	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → 0 ∈ ℤ)
162		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0)))
163	159, 160, 161, 162	fvdifsupp 8193	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → (𝐺‘𝑤) = 0)
164	163	oveq1d 7437	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = (0(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))
165		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
166	165	crngmgp 20474	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
167	6, 166	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
168	167	cmnmndd 20025	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
169	168	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → (mulGrp‘𝑅) ∈ Mnd)
170	17	subrgss 20831	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝑅) → 𝐹 ⊆ 𝐵)
171	3, 170	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
172	51, 171	unssd 4189	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ⊆ 𝐵)
173		sswrd 14672	. . . . . . . . . . . . . . 15 ⊢ ((𝐸 ∪ 𝐹) ⊆ 𝐵 → Word (𝐸 ∪ 𝐹) ⊆ Word 𝐵)
174	172, 173	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Word (𝐸 ∪ 𝐹) ⊆ Word 𝐵)
175	174	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → Word (𝐸 ∪ 𝐹) ⊆ Word 𝐵)
176	175	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → Word (𝐸 ∪ 𝐹) ⊆ Word 𝐵)
177	162	eldifad 4050	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → 𝑤 ∈ Word (𝐸 ∪ 𝐹))
178	176, 177	sseldd 3910	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → 𝑤 ∈ Word 𝐵)
179	165, 17	mgpbas 20372	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
180	179	gsumwcl 19042	. . . . . . . . . . 11 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝑤 ∈ Word 𝐵) → ((mulGrp‘𝑅) Σg 𝑤) ∈ 𝐵)
181	169, 178, 180	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → ((mulGrp‘𝑅) Σg 𝑤) ∈ 𝐵)
182	17, 5, 18	mulg0 19291	. . . . . . . . . 10 ⊢ (((mulGrp‘𝑅) Σg 𝑤) ∈ 𝐵 → (0(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 )
183	181, 182	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → (0(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 )
184	164, 183	eqtrd 2795	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0))) → ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 )
185	79	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → (𝐺 supp 0) ∈ Fin)
186	19	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → 𝑅 ∈ Grp)
187	34	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → 𝐺:Word (𝐸 ∪ 𝐹)⟶ℤ)
188	187	ffvelcdmda 7087	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → (𝐺‘𝑤) ∈ ℤ)
189	168	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → (mulGrp‘𝑅) ∈ Mnd)
190	175	sselda 3909	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → 𝑤 ∈ Word 𝐵)
191	189, 190, 180	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → ((mulGrp‘𝑅) Σg 𝑤) ∈ 𝐵)
192	17, 18, 186, 188, 191	mulgcld 19313	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) ∈ 𝐵)
193	117, 118	sseqtrid 3951	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 supp 0) ⊆ Word (𝐸 ∪ 𝐹))
194	193	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → (𝐺 supp 0) ⊆ Word (𝐸 ∪ 𝐹))
195	17, 5, 157, 158, 184, 185, 192, 194	gsummptres2 33662	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → (𝑅 Σg (𝑤 ∈ Word (𝐸 ∪ 𝐹) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))) = (𝑅 Σg (𝑤 ∈ (𝐺 supp 0) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))
196	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → 𝐹 ∈ (SubRing‘𝑅))
197	19	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → 𝑅 ∈ Grp)
198	34	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → 𝐺:Word (𝐸 ∪ 𝐹)⟶ℤ)
199	194	sselda 3909	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → 𝑤 ∈ Word (𝐸 ∪ 𝐹))
200	198, 199	ffvelcdmd 7088	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → (𝐺‘𝑤) ∈ ℤ)
201	168	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → (mulGrp‘𝑅) ∈ Mnd)
202	194, 175	sstrd 3919	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → (𝐺 supp 0) ⊆ Word 𝐵)
203	202	sselda 3909	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → 𝑤 ∈ Word 𝐵)
204	201, 203, 180	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → ((mulGrp‘𝑅) Σg 𝑤) ∈ 𝐵)
205	17, 18, 197, 200, 204	mulgcld 19313	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) ∈ 𝐵)
206	26	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → 𝑞:Word (𝐸 ∪ 𝐹)⟶(𝐸 × 𝐹))
207	206, 199	ffvelcdmd 7088	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → (𝑞‘𝑤) ∈ (𝐸 × 𝐹))
208		xp2nd 8036	. . . . . . . . 9 ⊢ ((𝑞‘𝑤) ∈ (𝐸 × 𝐹) → (2nd ‘(𝑞‘𝑤)) ∈ 𝐹)
209	207, 208	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑤 ∈ (𝐺 supp 0)) → (2nd ‘(𝑞‘𝑤)) ∈ 𝐹)
210		2fveq3 6893	. . . . . . . . 9 ⊢ (𝑣 = 𝑤 → (2nd ‘(𝑞‘𝑣)) = (2nd ‘(𝑞‘𝑤)))
211	210	cbvmptv 5208	. . . . . . . 8 ⊢ (𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) = (𝑤 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑤)))
212	17, 5, 157, 185, 196, 205, 209, 211	gsummpt2co 33657	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → (𝑅 Σg (𝑤 ∈ (𝐺 supp 0) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))) = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))))
213	195, 212	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → (𝑅 Σg (𝑤 ∈ Word (𝐸 ∪ 𝐹) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))) = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))))
214	213	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → (𝑅 Σg (𝑤 ∈ Word (𝐸 ∪ 𝐹) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))) = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))))
215		elrgspnsubrunlem2.3	. . . . . 6 ⊢ (𝜑 → 𝑋 = (𝑅 Σg (𝑤 ∈ Word (𝐸 ∪ 𝐹) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))
216	215	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → 𝑋 = (𝑅 Σg (𝑤 ∈ Word (𝐸 ∪ 𝐹) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))
217	7	ad4antr 745	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑅 ∈ Ring)
218	51	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝐸 ⊆ 𝐵)
219	26	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑞:Word (𝐸 ∪ 𝐹)⟶(𝐸 × 𝐹))
220	135	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ Word (𝐸 ∪ 𝐹))
221	220	sselda 3909	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑣 ∈ Word (𝐸 ∪ 𝐹))
222	219, 221	ffvelcdmd 7088	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝑞‘𝑣) ∈ (𝐸 × 𝐹))
223	222, 139	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (1st ‘(𝑞‘𝑣)) ∈ 𝐸)
224	218, 223	sseldd 3910	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (1st ‘(𝑞‘𝑣)) ∈ 𝐵)
225	224	adantllr 732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (1st ‘(𝑞‘𝑣)) ∈ 𝐵)
226	196, 170	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → 𝐹 ⊆ 𝐵)
227	226	sselda 3909	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → 𝑓 ∈ 𝐵)
228	227	ad4ant13 764	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑓 ∈ 𝐵)
229		elrgspnsubrun.t	. . . . . . . . . . . . . 14 ⊢ · = (.r‘𝑅)
230	17, 18, 229	mulgass2 20547	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ ((𝐺‘𝑣) ∈ ℤ ∧ (1st ‘(𝑞‘𝑣)) ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) · 𝑓) = ((𝐺‘𝑣)(.g‘𝑅)((1st ‘(𝑞‘𝑣)) · 𝑓)))
231	217, 41, 225, 228, 230	syl13anc 1399	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) · 𝑓) = ((𝐺‘𝑣)(.g‘𝑅)((1st ‘(𝑞‘𝑣)) · 𝑓)))
232		oveq2 7430	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑣 → ((mulGrp‘𝑅) Σg 𝑤) = ((mulGrp‘𝑅) Σg 𝑣))
233		2fveq3 6893	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑣 → (1st ‘(𝑞‘𝑤)) = (1st ‘(𝑞‘𝑣)))
234		2fveq3 6893	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑣 → (2nd ‘(𝑞‘𝑤)) = (2nd ‘(𝑞‘𝑣)))
235	233, 234	oveq12d 7440	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑣 → ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤))) = ((1st ‘(𝑞‘𝑣)) · (2nd ‘(𝑞‘𝑣))))
236	232, 235	eqeq12d 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑣 → (((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤))) ↔ ((mulGrp‘𝑅) Σg 𝑣) = ((1st ‘(𝑞‘𝑣)) · (2nd ‘(𝑞‘𝑣)))))
237		simpllr 788	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤))))
238	236, 237, 40	rspcdva 3577	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((mulGrp‘𝑅) Σg 𝑣) = ((1st ‘(𝑞‘𝑣)) · (2nd ‘(𝑞‘𝑣))))
239	26	ffnd 6713	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) → 𝑞 Fn Word (𝐸 ∪ 𝐹))
240	239	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑞 Fn Word (𝐸 ∪ 𝐹))
241		elpreima 7060	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 Fn Word (𝐸 ∪ 𝐹) → (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↔ (𝑣 ∈ Word (𝐸 ∪ 𝐹) ∧ (𝑞‘𝑣) ∈ (𝐸 × {𝑓}))))
242	241	simplbda 505	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑞 Fn Word (𝐸 ∪ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝑞‘𝑣) ∈ (𝐸 × {𝑓}))
243	240, 242	sylancom 600	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝑞‘𝑣) ∈ (𝐸 × {𝑓}))
244		xp2nd 8036	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞‘𝑣) ∈ (𝐸 × {𝑓}) → (2nd ‘(𝑞‘𝑣)) ∈ {𝑓})
245	243, 244	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (2nd ‘(𝑞‘𝑣)) ∈ {𝑓})
246	245	elsnd 4598	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (2nd ‘(𝑞‘𝑣)) = 𝑓)
247	246	adantllr 732	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (2nd ‘(𝑞‘𝑣)) = 𝑓)
248	247	oveq2d 7438	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((1st ‘(𝑞‘𝑣)) · (2nd ‘(𝑞‘𝑣))) = ((1st ‘(𝑞‘𝑣)) · 𝑓))
249	238, 248	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((mulGrp‘𝑅) Σg 𝑣) = ((1st ‘(𝑞‘𝑣)) · 𝑓))
250	249	oveq2d 7438	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = ((𝐺‘𝑣)(.g‘𝑅)((1st ‘(𝑞‘𝑣)) · 𝑓)))
251	231, 250	eqtr4d 2798	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) · 𝑓) = ((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)))
252	251	mpteq2dva 5197	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ (((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) · 𝑓)) = (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣))))
253		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (𝐺‘𝑣) = (𝐺‘𝑤))
254		oveq2 7430	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → ((mulGrp‘𝑅) Σg 𝑣) = ((mulGrp‘𝑅) Σg 𝑤))
255	253, 254	oveq12d 7440	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑤 → ((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))
256	255	cbvmptv 5208	. . . . . . . . . 10 ⊢ (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣))) = (𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))
257	252, 256	eqtrdi 2811	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ (((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) · 𝑓)) = (𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤))))
258	257	oveq2d 7438	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ (((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) · 𝑓))) = (𝑅 Σg (𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))
259	7	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → 𝑅 ∈ Ring)
260	12	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (◡𝑞 “ (𝐸 × {𝑓})) ∈ V)
261	19	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑅 ∈ Grp)
262	187	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝐺:Word (𝐸 ∪ 𝐹)⟶ℤ)
263	262, 221	ffvelcdmd 7088	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝐺‘𝑣) ∈ ℤ)
264	17, 18, 261, 263, 224	mulgcld 19313	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) ∈ 𝐵)
265	46	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ Word (𝐸 ∪ 𝐹) ↦ (𝐺‘𝑣)) finSupp 0)
266		0zd 12710	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → 0 ∈ ℤ)
267	265, 220, 266	fmptssfisupp 9391	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ (𝐺‘𝑣)) finSupp 0)
268	54	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑦 ∈ 𝐵) → (0(.g‘𝑅)𝑦) = 0 )
269	56	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → 0 ∈ V)
270	267, 268, 263, 224, 269	fsuppssov1 9381	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))) finSupp 0 )
271	17, 5, 229, 259, 260, 227, 264, 270	gsummulc1 20552	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ (((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) · 𝑓))) = ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓))
272	271	adantlr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ (((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))) · 𝑓))) = ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓))
273	157	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → 𝑅 ∈ CMnd)
274	85	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → 𝐺 Fn Word (𝐸 ∪ 𝐹))
275	158	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → Word (𝐸 ∪ 𝐹) ∈ V)
276		0zd 12710	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → 0 ∈ ℤ)
277	135	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ Word (𝐸 ∪ 𝐹))
278		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})))
279	278	eldifad 4050	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})))
280	277, 279	sseldd 3910	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → 𝑣 ∈ Word (𝐸 ∪ 𝐹))
281		eldif 4048	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) ↔ (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ∧ ¬ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})))
282		nfv 1946	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑢(((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (𝐺 supp 0))
283		fvexd 6903	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (𝐺 supp 0)) ∧ 𝑢 ∈ (𝐺 supp 0)) → (2nd ‘(𝑞‘𝑢)) ∈ V)
284		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) = (𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢)))
285	282, 283, 284	fnmptd 6683	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (𝐺 supp 0)) → (𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) Fn (𝐺 supp 0))
286	285	adantlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ 𝑣 ∈ (𝐺 supp 0)) → (𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) Fn (𝐺 supp 0))
287		simpr 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ 𝑣 ∈ (𝐺 supp 0)) → 𝑣 ∈ (𝐺 supp 0))
288		2fveq3 6893	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑣 → (2nd ‘(𝑞‘𝑢)) = (2nd ‘(𝑞‘𝑣)))
289		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (𝐺 supp 0)) → 𝑣 ∈ (𝐺 supp 0))
290		fvexd 6903	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (𝐺 supp 0)) → (2nd ‘(𝑞‘𝑣)) ∈ V)
291	284, 288, 289, 290	fvmptd3 7020	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (𝐺 supp 0)) → ((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢)))‘𝑣) = (2nd ‘(𝑞‘𝑣)))
292	291	adantlr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ 𝑣 ∈ (𝐺 supp 0)) → ((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢)))‘𝑣) = (2nd ‘(𝑞‘𝑣)))
293	239	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ 𝑣 ∈ (𝐺 supp 0)) → 𝑞 Fn Word (𝐸 ∪ 𝐹))
294		simplr 781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ 𝑣 ∈ (𝐺 supp 0)) → 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})))
295	293, 294, 242	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ 𝑣 ∈ (𝐺 supp 0)) → (𝑞‘𝑣) ∈ (𝐸 × {𝑓}))
296	295, 244	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ 𝑣 ∈ (𝐺 supp 0)) → (2nd ‘(𝑞‘𝑣)) ∈ {𝑓})
297	292, 296	eqeltrd 2860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ 𝑣 ∈ (𝐺 supp 0)) → ((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢)))‘𝑣) ∈ {𝑓})
298	286, 287, 297	elpreimad 7061	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ 𝑣 ∈ (𝐺 supp 0)) → 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))
299	298	stoic1a 1804	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) ∧ ¬ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → ¬ 𝑣 ∈ (𝐺 supp 0))
300	299	anasss 472	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ∧ ¬ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → ¬ 𝑣 ∈ (𝐺 supp 0))
301	281, 300	sylan2b 606	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → ¬ 𝑣 ∈ (𝐺 supp 0))
302	280, 301	eldifd 4049	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → 𝑣 ∈ (Word (𝐸 ∪ 𝐹) ∖ (𝐺 supp 0)))
303	274, 275, 276, 302	fvdifsupp 8193	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → (𝐺‘𝑣) = 0)
304	303	oveq1d 7437	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → ((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = (0(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)))
305	168	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → (mulGrp‘𝑅) ∈ Mnd)
306	175	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → Word (𝐸 ∪ 𝐹) ⊆ Word 𝐵)
307	220, 306	sstrd 3919	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (◡𝑞 “ (𝐸 × {𝑓})) ⊆ Word 𝐵)
308	307	ssdifssd 4082	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) ⊆ Word 𝐵)
309	308	sselda 3909	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → 𝑣 ∈ Word 𝐵)
310	179	gsumwcl 19042	. . . . . . . . . . . . . . . 16 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝑣 ∈ Word 𝐵) → ((mulGrp‘𝑅) Σg 𝑣) ∈ 𝐵)
311	305, 309, 310	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → ((mulGrp‘𝑅) Σg 𝑣) ∈ 𝐵)
312	17, 5, 18	mulg0 19291	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝑅) Σg 𝑣) ∈ 𝐵 → (0(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = 0 )
313	311, 312	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → (0(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = 0 )
314	304, 313	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))) → ((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = 0 )
315	314	ralrimiva 3154	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → ∀𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = 0 )
316	255	eqeq1d 2762	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑤 → (((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = 0 ↔ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 ))
317	316	cbvralvw 3240	. . . . . . . . . . . . 13 ⊢ (∀𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = 0 ↔ ∀𝑤 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 )
318		2fveq3 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑤 → (2nd ‘(𝑞‘𝑢)) = (2nd ‘(𝑞‘𝑤)))
319	318	cbvmptv 5208	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) = (𝑤 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑤)))
320	319, 211	eqtr4i 2786	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) = (𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣)))
321	320	cnveqi 5852	. . . . . . . . . . . . . . . 16 ⊢ ◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) = ◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣)))
322	321	imaeq1i 6050	. . . . . . . . . . . . . . 15 ⊢ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}) = (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓})
323	322	difeq2i 4059	. . . . . . . . . . . . . 14 ⊢ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) = ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}))
324	323	raleqi 3317	. . . . . . . . . . . . 13 ⊢ (∀𝑤 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 ↔ ∀𝑤 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}))((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 )
325	317, 324	bitri 278	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))((𝐺‘𝑣)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑣)) = 0 ↔ ∀𝑤 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}))((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 )
326	315, 325	sylib 221	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → ∀𝑤 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}))((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 )
327	326	r19.21bi 3254	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑤 ∈ ((◡𝑞 “ (𝐸 × {𝑓})) ∖ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}))) → ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) = 0 )
328	185	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (𝐺 supp 0) ∈ Fin)
329	328	cnvimamptfin 9347	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ∈ Fin)
330	19	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑅 ∈ Grp)
331	187	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝐺:Word (𝐸 ∪ 𝐹)⟶ℤ)
332	220	sselda 3909	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑤 ∈ Word (𝐸 ∪ 𝐹))
333	331, 332	ffvelcdmd 7088	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (𝐺‘𝑤) ∈ ℤ)
334	168	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → (mulGrp‘𝑅) ∈ Mnd)
335	307	sselda 3909	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → 𝑤 ∈ Word 𝐵)
336	334, 335, 180	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((mulGrp‘𝑅) Σg 𝑤) ∈ 𝐵)
337	17, 18, 330, 333, 336	mulgcld 19313	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓}))) → ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)) ∈ 𝐵)
338	239	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → 𝑞 Fn Word (𝐸 ∪ 𝐹))
339	194	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → (𝐺 supp 0) ⊆ Word (𝐸 ∪ 𝐹))
340		nfv 1946	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤(((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}))
341		fvexd 6903	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) ∧ 𝑤 ∈ (𝐺 supp 0)) → (2nd ‘(𝑞‘𝑤)) ∈ V)
342	340, 341, 319	fnmptd 6683	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → (𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) Fn (𝐺 supp 0))
343		elpreima 7060	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) Fn (𝐺 supp 0) → (𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}) ↔ (𝑣 ∈ (𝐺 supp 0) ∧ ((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢)))‘𝑣) ∈ {𝑓})))
344	343	simprbda 504	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) Fn (𝐺 supp 0) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → 𝑣 ∈ (𝐺 supp 0))
345	342, 344	sylancom 600	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → 𝑣 ∈ (𝐺 supp 0))
346	339, 345	sseldd 3910	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → 𝑣 ∈ Word (𝐸 ∪ 𝐹))
347	26	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → 𝑞:Word (𝐸 ∪ 𝐹)⟶(𝐸 × 𝐹))
348	347, 346	ffvelcdmd 7088	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → (𝑞‘𝑣) ∈ (𝐸 × 𝐹))
349		1st2nd2 8042	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞‘𝑣) ∈ (𝐸 × 𝐹) → (𝑞‘𝑣) = ⟨(1st ‘(𝑞‘𝑣)), (2nd ‘(𝑞‘𝑣))⟩)
350	348, 349	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → (𝑞‘𝑣) = ⟨(1st ‘(𝑞‘𝑣)), (2nd ‘(𝑞‘𝑣))⟩)
351	348, 139	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → (1st ‘(𝑞‘𝑣)) ∈ 𝐸)
352	345, 291	syldan 603	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → ((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢)))‘𝑣) = (2nd ‘(𝑞‘𝑣)))
353	343	simplbda 505	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) Fn (𝐺 supp 0) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → ((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢)))‘𝑣) ∈ {𝑓})
354	342, 353	sylancom 600	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → ((𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢)))‘𝑣) ∈ {𝑓})
355	352, 354	eqeltrrd 2861	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → (2nd ‘(𝑞‘𝑣)) ∈ {𝑓})
356	351, 355	opelxpd 5689	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → ⟨(1st ‘(𝑞‘𝑣)), (2nd ‘(𝑞‘𝑣))⟩ ∈ (𝐸 × {𝑓}))
357	350, 356	eqeltrd 2860	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → (𝑞‘𝑣) ∈ (𝐸 × {𝑓}))
358	338, 346, 357	elpreimad 7061	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) ∧ 𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓})) → 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})))
359	358	ex 418	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (𝑣 ∈ (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}) → 𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓}))))
360	359	ssrdv 3915	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (◡(𝑢 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑢))) “ {𝑓}) ⊆ (◡𝑞 “ (𝐸 × {𝑓})))
361	322, 360	eqsstrrid 3948	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ⊆ (◡𝑞 “ (𝐸 × {𝑓})))
362	17, 5, 273, 260, 327, 329, 337, 361	gsummptres2 33662	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ 𝑓 ∈ 𝐹) → (𝑅 Σg (𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))) = (𝑅 Σg (𝑤 ∈ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))
363	362	adantlr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → (𝑅 Σg (𝑤 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))) = (𝑅 Σg (𝑤 ∈ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))
364	258, 272, 363	3eqtr3d 2803	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) ∧ 𝑓 ∈ 𝐹) → ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓) = (𝑅 Σg (𝑤 ∈ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))
365	364	mpteq2dva 5197	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → (𝑓 ∈ 𝐹 ↦ ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓)) = (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤))))))
366	365	oveq2d 7438	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓))) = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑤 ∈ (◡(𝑣 ∈ (𝐺 supp 0) ↦ (2nd ‘(𝑞‘𝑣))) “ {𝑓}) ↦ ((𝐺‘𝑤)(.g‘𝑅)((mulGrp‘𝑅) Σg 𝑤)))))))
367	214, 216, 366	3eqtr4d 2805	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓))))
368	155, 367	jca 521	. . 3 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → ((𝑓 ∈ 𝐹 ↦ (𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣)))))) finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑅 Σg (𝑣 ∈ (◡𝑞 “ (𝐸 × {𝑓})) ↦ ((𝐺‘𝑣)(.g‘𝑅)(1st ‘(𝑞‘𝑣))))) · 𝑓)))))
369	61, 75, 368	rspcedvd 3578	. 2 ⊢ (((𝜑 ∧ 𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))) ∧ ∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))) → ∃𝑝 ∈ (𝐸 ↑m 𝐹)(𝑝 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑝‘𝑓) · 𝑓)))))
370		fveq2 6888	. . . . 5 ⊢ (𝑎 = (𝑞‘𝑤) → (1st ‘𝑎) = (1st ‘(𝑞‘𝑤)))
371		fveq2 6888	. . . . 5 ⊢ (𝑎 = (𝑞‘𝑤) → (2nd ‘𝑎) = (2nd ‘(𝑞‘𝑤)))
372	370, 371	oveq12d 7440	. . . 4 ⊢ (𝑎 = (𝑞‘𝑤) → ((1st ‘𝑎) · (2nd ‘𝑎)) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤))))
373	372	eqeq2d 2771	. . 3 ⊢ (𝑎 = (𝑞‘𝑤) → (((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘𝑎) · (2nd ‘𝑎)) ↔ ((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤)))))
374		vex 3454	. . . . . . . 8 ⊢ 𝑒 ∈ V
375		vex 3454	. . . . . . . 8 ⊢ 𝑓 ∈ V
376	374, 375	op1std 8013	. . . . . . 7 ⊢ (𝑎 = ⟨𝑒, 𝑓⟩ → (1st ‘𝑎) = 𝑒)
377	374, 375	op2ndd 8014	. . . . . . 7 ⊢ (𝑎 = ⟨𝑒, 𝑓⟩ → (2nd ‘𝑎) = 𝑓)
378	376, 377	oveq12d 7440	. . . . . 6 ⊢ (𝑎 = ⟨𝑒, 𝑓⟩ → ((1st ‘𝑎) · (2nd ‘𝑎)) = (𝑒 · 𝑓))
379	378	eqeq2d 2771	. . . . 5 ⊢ (𝑎 = ⟨𝑒, 𝑓⟩ → (((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘𝑎) · (2nd ‘𝑎)) ↔ ((mulGrp‘𝑅) Σg 𝑤) = (𝑒 · 𝑓)))
380		simpllr 788	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ ((mulGrp‘𝑅) Σg 𝑤) = (𝑒 · 𝑓)) → 𝑒 ∈ 𝐸)
381		simplr 781	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ ((mulGrp‘𝑅) Σg 𝑤) = (𝑒 · 𝑓)) → 𝑓 ∈ 𝐹)
382	380, 381	opelxpd 5689	. . . . 5 ⊢ (((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ ((mulGrp‘𝑅) Σg 𝑤) = (𝑒 · 𝑓)) → ⟨𝑒, 𝑓⟩ ∈ (𝐸 × 𝐹))
383		simpr 490	. . . . 5 ⊢ (((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ ((mulGrp‘𝑅) Σg 𝑤) = (𝑒 · 𝑓)) → ((mulGrp‘𝑅) Σg 𝑤) = (𝑒 · 𝑓))
384	379, 382, 383	rspcedvdw 3579	. . . 4 ⊢ (((((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) ∧ 𝑒 ∈ 𝐸) ∧ 𝑓 ∈ 𝐹) ∧ ((mulGrp‘𝑅) Σg 𝑤) = (𝑒 · 𝑓)) → ∃𝑎 ∈ (𝐸 × 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘𝑎) · (2nd ‘𝑎)))
385	165, 229	mgpplusg 20371	. . . . 5 ⊢ · = (+g‘(mulGrp‘𝑅))
386	167	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → (mulGrp‘𝑅) ∈ CMnd)
387	165	subrgsubm 20844	. . . . . . 7 ⊢ (𝐸 ∈ (SubRing‘𝑅) → 𝐸 ∈ (SubMnd‘(mulGrp‘𝑅)))
388	1, 387	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (SubMnd‘(mulGrp‘𝑅)))
389	388	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → 𝐸 ∈ (SubMnd‘(mulGrp‘𝑅)))
390	165	subrgsubm 20844	. . . . . . 7 ⊢ (𝐹 ∈ (SubRing‘𝑅) → 𝐹 ∈ (SubMnd‘(mulGrp‘𝑅)))
391	3, 390	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubMnd‘(mulGrp‘𝑅)))
392	391	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → 𝐹 ∈ (SubMnd‘(mulGrp‘𝑅)))
393		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → 𝑤 ∈ Word (𝐸 ∪ 𝐹))
394	385, 386, 389, 392, 393	gsumwun 33685	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → ∃𝑒 ∈ 𝐸 ∃𝑓 ∈ 𝐹 ((mulGrp‘𝑅) Σg 𝑤) = (𝑒 · 𝑓))
395	384, 394	r19.29vva 3222	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ Word (𝐸 ∪ 𝐹)) → ∃𝑎 ∈ (𝐸 × 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘𝑎) · (2nd ‘𝑎)))
396	373, 24, 21, 395	ac6mapd 33266	. 2 ⊢ (𝜑 → ∃𝑞 ∈ ((𝐸 × 𝐹) ↑m Word (𝐸 ∪ 𝐹))∀𝑤 ∈ Word (𝐸 ∪ 𝐹)((mulGrp‘𝑅) Σg 𝑤) = ((1st ‘(𝑞‘𝑤)) · (2nd ‘(𝑞‘𝑤))))
397	369, 396	r19.29a 3170	1 ⊢ (𝜑 → ∃𝑝 ∈ (𝐸 ↑m 𝐹)(𝑝 finSupp 0 ∧ 𝑋 = (𝑅 Σg (𝑓 ∈ 𝐹 ↦ ((𝑝‘𝑓) · 𝑓)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088   ∪ cun 4148  {csn 4584  ⟨cop 4834   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   “ cima 5653  Rel wrel 5655  Fun wfun 6535   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422  1^st c1st 8001  2^nd c2nd 8002   supp csupp 8181   ↑_m cmap 8852  Fincfn 8978   finSupp cfsupp 9358  0cc0 11205  ℤcz 12698  Word cword 14663  Basecbs 17394  ._rcmulr 17436  0_gc0g 17617   Σ_g cgsu 17618  Mndcmnd 18930  SubMndcsubmnd 18984  Grpcgrp 19151  ._gcmg 19284  SubGrpcsubg 19337  CMndccmn 20001  Abelcabl 20002  mulGrpcmgp 20367  Ringcrg 20466  CRingccrg 20467  SubRingcsubrg 20828  RingSpancrgspn 20869

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-ac2 10546  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-of 7695  df-om 7880  df-1st 8003  df-2nd 8004  df-supp 8182  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-2o 8482  df-er 8722  df-map 8854  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-fsupp 9359  df-oi 9509  df-card 10025  df-ac 10200  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-3 12411  df-n0 12612  df-xnn0 12685  df-z 12699  df-uz 12971  df-fz 13645  df-fzo 13794  df-seq 14150  df-hash 14480  df-word 14664  df-lsw 14713  df-concat 14721  df-s1 14748  df-substr 14795  df-pfx 14827  df-sets 17349  df-slot 17367  df-ndx 17379  df-base 17395  df-ress 17416  df-plusg 17448  df-mulr 17449  df-0g 17619  df-gsum 17620  df-mre 17763  df-mrc 17764  df-acs 17766  df-mgm 18823  df-sgrp 18915  df-mnd 18931  df-mhm 18985  df-submnd 18986  df-grp 19154  df-minusg 19155  df-mulg 19285  df-subg 19340  df-ghm 19435  df-cntz 19538  df-cmn 20003  df-abl 20004  df-mgp 20368  df-rng 20382  df-ur 20415  df-ring 20468  df-cring 20469  df-subrg 20829

This theorem is used by:  elrgspnsubrun  33858