Theorem mnringmulrcld 40936

Description: Monoid rings are closed under multiplication. (Contributed by Rohan Ridenour, 14-May-2024.)

Hypotheses

Ref	Expression
mnringmulrcld.2	⊢ 𝐹 = (𝑅 MndRing 𝑀)
mnringmulrcld.3	⊢ 𝐵 = (Base‘𝐹)
mnringmulrcld.1	⊢ 𝐴 = (Base‘𝑀)
mnringmulrcld.4	⊢ · = (.r‘𝐹)
mnringmulrcld.5	⊢ (𝜑 → 𝑅 ∈ Ring)
mnringmulrcld.6	⊢ (𝜑 → 𝑀 ∈ 𝑈)
mnringmulrcld.7	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mnringmulrcld.8	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
mnringmulrcld	⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)

Proof of Theorem mnringmulrcld

Dummy variables 𝑎 𝑏 𝑝 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnringmulrcld.2	. . 3 ⊢ 𝐹 = (𝑅 MndRing 𝑀)
2		mnringmulrcld.3	. . 3 ⊢ 𝐵 = (Base‘𝐹)
3		eqid 2798	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2798	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		mnringmulrcld.1	. . 3 ⊢ 𝐴 = (Base‘𝑀)
6		eqid 2798	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
7		mnringmulrcld.4	. . 3 ⊢ · = (.r‘𝐹)
8		mnringmulrcld.5	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		mnringmulrcld.6	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
10		mnringmulrcld.7	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		mnringmulrcld.8	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	mnringmulrvald 40935	. 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))))
13		eqid 2798	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
14	1, 8, 9	mnringlmodd 40934	. . . 4 ⊢ (𝜑 → 𝐹 ∈ LMod)
15		lmodcmn 19675	. . . 4 ⊢ (𝐹 ∈ LMod → 𝐹 ∈ CMnd)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → 𝐹 ∈ CMnd)
17	5	fvexi 6659	. . . . 5 ⊢ 𝐴 ∈ V
18	17, 17	xpex 7456	. . . 4 ⊢ (𝐴 × 𝐴) ∈ V
19	18	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 × 𝐴) ∈ V)
20	8	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑅 ∈ Ring)
21		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘𝑅)
22	1, 2, 5, 21, 8, 9, 10	mnringbasefd 40926	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋:𝐴⟶(Base‘𝑅))
23	22	3ad2ant1 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑋:𝐴⟶(Base‘𝑅))
24		simp2 1134	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑎 ∈ 𝐴)
25	23, 24	ffvelrnd 6829	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑋‘𝑎) ∈ (Base‘𝑅))
26	1, 2, 5, 21, 8, 9, 11	mnringbasefd 40926	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌:𝐴⟶(Base‘𝑅))
27	26	3ad2ant1 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑌:𝐴⟶(Base‘𝑅))
28		simp3 1135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
29	27, 28	ffvelrnd 6829	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑌‘𝑏) ∈ (Base‘𝑅))
30	21, 3	ringcl 19307	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑎) ∈ (Base‘𝑅) ∧ (𝑌‘𝑏) ∈ (Base‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) ∈ (Base‘𝑅))
31	20, 25, 29, 30	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) ∈ (Base‘𝑅))
32	21, 4	ring0cl 19315	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
33	20, 32	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (0g‘𝑅) ∈ (Base‘𝑅))
34	31, 33	ifcld 4470	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)) ∈ (Base‘𝑅))
35	34	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑖 ∈ 𝐴) → if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)) ∈ (Base‘𝑅))
36	35	fmpttd 6856	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))):𝐴⟶(Base‘𝑅))
37	21	fvexi 6659	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
38	37, 17	elmap 8418	. . . . . . . . 9 ⊢ ((𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴) ↔ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))):𝐴⟶(Base‘𝑅))
39	36, 38	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴))
40	17	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝐴 ∈ V)
41		eqid 2798	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))
42	40, 33, 41	sniffsupp 8857	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) finSupp (0g‘𝑅))
43	39, 42	jca 515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴) ∧ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) finSupp (0g‘𝑅)))
44	9	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑀 ∈ 𝑈)
45	1, 2, 5, 21, 4, 20, 44	mnringelbased 40925	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵 ↔ ((𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴) ∧ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) finSupp (0g‘𝑅))))
46	43, 45	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵)
47	46	3expb 1117	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵)
48	47	ralrimivva 3156	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵)
49		eqid 2798	. . . . 5 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
50	49	fmpo 7748	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵 ↔ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))):(𝐴 × 𝐴)⟶𝐵)
51	48, 50	sylib 221	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))):(𝐴 × 𝐴)⟶𝐵)
52	17, 17	mpoex 7760	. . . . 5 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) ∈ V
53	52	a1i 11	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) ∈ V)
54	51	ffnd 6488	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) Fn (𝐴 × 𝐴))
55	13	fvexi 6659	. . . . 5 ⊢ (0g‘𝐹) ∈ V
56	55	a1i 11	. . . 4 ⊢ (𝜑 → (0g‘𝐹) ∈ V)
57	1, 2, 4, 8, 9, 10	mnringbasefsuppd 40927	. . . . . 6 ⊢ (𝜑 → 𝑋 finSupp (0g‘𝑅))
58	57	fsuppimpd 8824	. . . . 5 ⊢ (𝜑 → (𝑋 supp (0g‘𝑅)) ∈ Fin)
59	1, 2, 4, 8, 9, 11	mnringbasefsuppd 40927	. . . . . 6 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅))
60	59	fsuppimpd 8824	. . . . 5 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) ∈ Fin)
61		xpfi 8773	. . . . 5 ⊢ (((𝑋 supp (0g‘𝑅)) ∈ Fin ∧ (𝑌 supp (0g‘𝑅)) ∈ Fin) → ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∈ Fin)
62	58, 60, 61	syl2anc 587	. . . 4 ⊢ (𝜑 → ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∈ Fin)
63		elxpi 5541	. . . . . . 7 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)))
64		simpl 486	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑝 = ⟨𝑎, 𝑏⟩)
65	64	2eximi 1837	. . . . . . 7 ⊢ (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
66	63, 65	syl 17	. . . . . 6 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
67	66	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → ∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
68		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴))
69		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑎 𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅)))
70		nfmpo1 7213	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
71		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑎𝑝
72	70, 71	nffv 6655	. . . . . . . 8 ⊢ Ⅎ𝑎((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝)
73		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑎(0g‘𝐹)
74	72, 73	nfeq 2968	. . . . . . 7 ⊢ Ⅎ𝑎((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)
75	69, 74	nfor 1905	. . . . . 6 ⊢ Ⅎ𝑎(𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))
76		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑏(𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴))
77		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑏 𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅)))
78		nfmpo2 7214	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
79		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎ𝑏𝑝
80	78, 79	nffv 6655	. . . . . . . . 9 ⊢ Ⅎ𝑏((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝)
81		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑏(0g‘𝐹)
82	80, 81	nfeq 2968	. . . . . . . 8 ⊢ Ⅎ𝑏((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)
83	77, 82	nfor 1905	. . . . . . 7 ⊢ Ⅎ𝑏(𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))
84		simp3 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
85		simp2 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 ∈ (𝐴 × 𝐴))
86	84, 85	eqeltrrd 2891	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴))
87		opelxp 5555	. . . . . . . . . 10 ⊢ (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴))
88	86, 87	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴))
89		ianor 979	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ↔ (¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
90	22	ffnd 6488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 Fn 𝐴)
91	17	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐴 ∈ V)
92	4	fvexi 6659	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0g‘𝑅) ∈ V
93	92	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
94		elsuppfn 7821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 Fn 𝐴 ∧ 𝐴 ∈ V ∧ (0g‘𝑅) ∈ V) → (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑋‘𝑎) ≠ (0g‘𝑅))))
95	90, 91, 93, 94	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑋‘𝑎) ≠ (0g‘𝑅))))
96	95	biimprd 251	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑎 ∈ 𝐴 ∧ (𝑋‘𝑎) ≠ (0g‘𝑅)) → 𝑎 ∈ (𝑋 supp (0g‘𝑅))))
97	96	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑎 ∈ 𝐴 ∧ (𝑋‘𝑎) ≠ (0g‘𝑅)) → 𝑎 ∈ (𝑋 supp (0g‘𝑅))))
98	24, 97	mpand 694	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑋‘𝑎) ≠ (0g‘𝑅) → 𝑎 ∈ (𝑋 supp (0g‘𝑅))))
99	98	necon1bd 3005	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) → (𝑋‘𝑎) = (0g‘𝑅)))
100	26	ffnd 6488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑌 Fn 𝐴)
101		elsuppfn 7821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑌 Fn 𝐴 ∧ 𝐴 ∈ V ∧ (0g‘𝑅) ∈ V) → (𝑏 ∈ (𝑌 supp (0g‘𝑅)) ↔ (𝑏 ∈ 𝐴 ∧ (𝑌‘𝑏) ≠ (0g‘𝑅))))
102	100, 91, 93, 101	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑏 ∈ (𝑌 supp (0g‘𝑅)) ↔ (𝑏 ∈ 𝐴 ∧ (𝑌‘𝑏) ≠ (0g‘𝑅))))
103	102	biimprd 251	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑏 ∈ 𝐴 ∧ (𝑌‘𝑏) ≠ (0g‘𝑅)) → 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
104	103	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑏 ∈ 𝐴 ∧ (𝑌‘𝑏) ≠ (0g‘𝑅)) → 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
105	28, 104	mpand 694	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑌‘𝑏) ≠ (0g‘𝑅) → 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
106	105	necon1bd 3005	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅)) → (𝑌‘𝑏) = (0g‘𝑅)))
107	99, 106	orim12d 962	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) → ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))))
108	107	imp 410	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ (¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))) → ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅)))
109	89, 108	sylan2b 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ¬ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))) → ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅)))
110		oveq1 7142	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋‘𝑎) = (0g‘𝑅) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = ((0g‘𝑅)(.r‘𝑅)(𝑌‘𝑏)))
111	21, 3, 4	ringlz 19333	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ (𝑌‘𝑏) ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
112	20, 29, 111	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((0g‘𝑅)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
113	110, 112	sylan9eqr 2855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ (𝑋‘𝑎) = (0g‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
114		oveq2 7143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌‘𝑏) = (0g‘𝑅) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = ((𝑋‘𝑎)(.r‘𝑅)(0g‘𝑅)))
115	21, 3, 4	ringrz 19334	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑎) ∈ (Base‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
116	20, 25, 115	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑋‘𝑎)(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
117	114, 116	sylan9eqr 2855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ (𝑌‘𝑏) = (0g‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
118	113, 117	jaodan 955	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
119	118	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) ∧ 𝑖 = (𝑎(+g‘𝑀)𝑏)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
120		eqidd 2799	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) ∧ ¬ 𝑖 = (𝑎(+g‘𝑀)𝑏)) → (0g‘𝑅) = (0g‘𝑅))
121	119, 120	ifeqda 4460	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)) = (0g‘𝑅))
122	121	mpteq2dv 5126	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)))
123		fconstmpt 5578	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 × {(0g‘𝑅)}) = (𝑖 ∈ 𝐴 ↦ (0g‘𝑅))
124	1, 4, 5, 8, 9	mnring0g2d 40930	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 × {(0g‘𝑅)}) = (0g‘𝐹))
125	123, 124	syl5eqr 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)) = (0g‘𝐹))
126	125	3ad2ant1 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)) = (0g‘𝐹))
127	126	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)) = (0g‘𝐹))
128	122, 127	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹))
129	109, 128	syldan 594	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ¬ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹))
130	129	ex 416	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
131	130	orrd 860	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ∨ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
132	131	3expb 1117	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ∨ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
133	132	3adant3 1129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ((𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ∨ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
134		eleq1 2877	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ↔ ⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅)))))
135		opelxp 5555	. . . . . . . . . . . . 13 ⊢ (⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
136	134, 135	syl6bb 290	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))))
137	136	3ad2ant3 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))))
138		simp2l 1196	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑎 ∈ 𝐴)
139		simp2r 1197	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑏 ∈ 𝐴)
140		eqidd 2799	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))))
141		simp3 1135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
142	17	mptex 6963	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ V
143	142	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ V)
144	140, 141, 143	fvmpopr2d 7290	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
145	138, 139, 144	mpd3an23 1460	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
146	145	eqeq1d 2800	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹) ↔ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
147	137, 146	orbi12d 916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ((𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)) ↔ ((𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ∨ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹))))
148	133, 147	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)))
149	88, 148	syld3an2 1408	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)))
150	149	3expia 1118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))))
151	76, 83, 150	exlimd 2216	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))))
152	68, 75, 151	exlimd 2216	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))))
153	67, 152	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)))
154	53, 54, 56, 62, 153	finnzfsuppd 40915	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) finSupp (0g‘𝐹))
155	2, 13, 16, 19, 51, 154	gsumcl 19028	. 2 ⊢ (𝜑 → (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))) ∈ 𝐵)
156	12, 155	eqeltrd 2890	1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441  ifcif 4425  {csn 4525  ⟨cop 4531   class class class wbr 5030   ↦ cmpt 5110   × cxp 5517   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   supp csupp 7813   ↑_m cmap 8389  Fincfn 8492   finSupp cfsupp 8817  Basecbs 16475  +_gcplusg 16557  ._rcmulr 16558  0_gc0g 16705   Σ_g cgsu 16706  CMndccmn 18898  Ringcrg 19290  LModclmod 19627   MndRing cmnring 40919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-sup 8890  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-gsum 16708  df-prds 16713  df-pws 16715  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19233  df-ur 19245  df-ring 19292  df-subrg 19526  df-lmod 19629  df-lss 19697  df-sra 19937  df-rgmod 19938  df-dsmm 20421  df-frlm 20436  df-mnring 40920

This theorem is referenced by: (None)