Theorem mnringmulrcld 45074

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 2762	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2762	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		mnringmulrcld.1	. . 3 ⊢ 𝐴 = (Base‘𝑀)
6		eqid 2762	. . 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 45073	. 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 2762	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
14	1, 8, 9	mnringlmodd 45072	. . . 4 ⊢ (𝜑 → 𝐹 ∈ LMod)
15		lmodcmn 21100	. . . 4 ⊢ (𝐹 ∈ LMod → 𝐹 ∈ CMnd)
16	14, 15	syl 18	. . 3 ⊢ (𝜑 → 𝐹 ∈ CMnd)
17	5	fvexi 6896	. . . . 5 ⊢ 𝐴 ∈ V
18	17, 17	xpex 7756	. . . 4 ⊢ (𝐴 × 𝐴) ∈ V
19	18	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 × 𝐴) ∈ V)
20	8	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑅 ∈ Ring)
21		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘𝑅)
22	1, 2, 5, 21, 8, 9, 10	mnringbasefd 45064	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋:𝐴⟶(Base‘𝑅))
23	22	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑋:𝐴⟶(Base‘𝑅))
24		simp2 1155	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑎 ∈ 𝐴)
25	23, 24	ffvelcdmd 7082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑋‘𝑎) ∈ (Base‘𝑅))
26	1, 2, 5, 21, 8, 9, 11	mnringbasefd 45064	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌:𝐴⟶(Base‘𝑅))
27	26	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑌:𝐴⟶(Base‘𝑅))
28		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
29	27, 28	ffvelcdmd 7082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑌‘𝑏) ∈ (Base‘𝑅))
30	21, 3	ringcl 20395	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) ∈ (Base‘𝑅))
32	21, 4	ring0cl 20414	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
33	20, 32	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (0g‘𝑅) ∈ (Base‘𝑅))
34	31, 33	ifcld 4532	. . . . . . . . . . 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 486	. . . . . . . . . 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 7112	. . . . . . . . 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 6896	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
38	37, 17	elmap 8882	. . . . . . . . 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 2762	. . . . . . . . 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 9374	. . . . . . . 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 521	. . . . . . 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 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑀 ∈ 𝑈)
45	1, 2, 5, 21, 4, 20, 44	mnringelbased 45063	. . . . . . 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 1138	. . . . 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 3207	. . . 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 2762	. . . . 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 8069	. . . 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 8082	. . . . 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 6707	. . . 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 6896	. . . . 5 ⊢ (0g‘𝐹) ∈ V
56	55	a1i 11	. . . 4 ⊢ (𝜑 → (0g‘𝐹) ∈ V)
57	1, 2, 4, 8, 9, 10	mnringbasefsuppd 45065	. . . . . 6 ⊢ (𝜑 → 𝑋 finSupp (0g‘𝑅))
58	57	fsuppimpd 9343	. . . . 5 ⊢ (𝜑 → (𝑋 supp (0g‘𝑅)) ∈ Fin)
59	1, 2, 4, 8, 9, 11	mnringbasefsuppd 45065	. . . . . 6 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅))
60	59	fsuppimpd 9343	. . . . 5 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) ∈ Fin)
61		xpfi 9293	. . . . 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 596	. . . 4 ⊢ (𝜑 → ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∈ Fin)
63		elxpi 5681	. . . . . . 7 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)))
64		simpl 488	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑝 = ⟨𝑎, 𝑏⟩)
65	64	2eximi 1869	. . . . . . 7 ⊢ (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
66	63, 65	syl 18	. . . . . 6 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
67	66	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → ∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
68		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴))
69		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑎 𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅)))
70		nfmpo1 7497	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
71		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑎𝑝
72	70, 71	nffv 6892	. . . . . . . 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 2924	. . . . . . . 8 ⊢ Ⅎ𝑎(0g‘𝐹)
74	72, 73	nfeq 2937	. . . . . . 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 1937	. . . . . 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 1947	. . . . . . 7 ⊢ Ⅎ𝑏(𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴))
77		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑏 𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅)))
78		nfmpo2 7498	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
79		nfcv 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑏𝑝
80	78, 79	nffv 6892	. . . . . . . . 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 2924	. . . . . . . . 9 ⊢ Ⅎ𝑏(0g‘𝐹)
82	80, 81	nfeq 2937	. . . . . . . 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 1937	. . . . . . 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 1156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
85		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 ∈ (𝐴 × 𝐴))
86	84, 85	eqeltrrd 2863	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴))
87		opelxp 5695	. . . . . . . . . 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 997	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ↔ (¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
90	22	ffnd 6707	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 Fn 𝐴)
91	17	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐴 ∈ V)
92	4	fvexi 6896	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0g‘𝑅) ∈ V
93	92	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
94		elsuppfn 8172	. . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑎 ∈ 𝐴 ∧ (𝑋‘𝑎) ≠ (0g‘𝑅)) → 𝑎 ∈ (𝑋 supp (0g‘𝑅))))
98	24, 97	mpand 708	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑋‘𝑎) ≠ (0g‘𝑅) → 𝑎 ∈ (𝑋 supp (0g‘𝑅))))
99	98	necon1bd 2975	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) → (𝑋‘𝑎) = (0g‘𝑅)))
100	26	ffnd 6707	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑌 Fn 𝐴)
101		elsuppfn 8172	. . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑏 ∈ 𝐴 ∧ (𝑌‘𝑏) ≠ (0g‘𝑅)) → 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
105	28, 104	mpand 708	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑌‘𝑏) ≠ (0g‘𝑅) → 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
106	105	necon1bd 2975	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅)) → (𝑌‘𝑏) = (0g‘𝑅)))
107	99, 106	orim12d 979	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . 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 606	. . . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋‘𝑎) = (0g‘𝑅) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = ((0g‘𝑅)(.r‘𝑅)(𝑌‘𝑏)))
111	21, 3, 4	ringlz 20441	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ (𝑌‘𝑏) ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
112	20, 29, 111	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((0g‘𝑅)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
113	110, 112	sylan9eqr 2819	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ (𝑋‘𝑎) = (0g‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
114		oveq2 7425	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌‘𝑏) = (0g‘𝑅) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = ((𝑋‘𝑎)(.r‘𝑅)(0g‘𝑅)))
115	21, 3, 4	ringrz 20442	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑎) ∈ (Base‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
116	20, 25, 115	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑋‘𝑎)(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
117	114, 116	sylan9eqr 2819	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ (𝑌‘𝑏) = (0g‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
118	113, 117	jaodan 972	. . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . 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 4522	. . . . . . . . . . . . . . . . 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 5203	. . . . . . . . . . . . . . . 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 5721	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 × {(0g‘𝑅)}) = (𝑖 ∈ 𝐴 ↦ (0g‘𝑅))
124	1, 4, 5, 8, 9	mnring0g2d 45068	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 × {(0g‘𝑅)}) = (0g‘𝐹))
125	123, 124	eqtr3id 2811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)) = (0g‘𝐹))
126	125	3ad2ant1 1151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)) = (0g‘𝐹))
127	126	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)) = (0g‘𝐹))
128	122, 127	eqtrd 2797	. . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 877	. . . . . . . . . . . 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 1138	. . . . . . . . . . 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 1150	. . . . . . . . . 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 2850	. . . . . . . . . . . . 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 5695	. . . . . . . . . . . . 13 ⊢ (⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
136	134, 135	bitrdi 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 1153	. . . . . . . . . . 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 1218	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑎 ∈ 𝐴)
139		simp2r 1219	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑏 ∈ 𝐴)
140		eqidd 2763	. . . . . . . . . . . . . 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 1156	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
142	17	mptex 7226	. . . . . . . . . . . . . . 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 7579	. . . . . . . . . . . . 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 1492	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 932	. . . . . . . . . 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 1438	. . . . . . . 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 1139	. . . . . . 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 2256	. . . . . 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 2256	. . . . 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 16	. . . 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 9347	. . 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 20048	. 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 2862	1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  Vcvv 3453  ifcif 4485  {csn 4587  ⟨cop 4593   class class class wbr 5107   ↦ cmpt 5190   × cxp 5657   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419   supp csupp 8162   ↑_m cmap 8830  Fincfn 8956   finSupp cfsupp 9335  Basecbs 17307  +_gcplusg 17348  ._rcmulr 17349  0_gc0g 17530   Σ_g cgsu 17531  CMndccmn 19913  Ringcrg 20378  LModclmod 21050   MndRing cmnring 45057

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-sup 9416  df-oi 9486  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-z 12620  df-dec 12741  df-uz 12892  df-fz 13566  df-fzo 13714  df-seq 14070  df-hash 14399  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-sca 17364  df-vsca 17365  df-ip 17366  df-tset 17367  df-ple 17368  df-ds 17370  df-hom 17372  df-cco 17373  df-0g 17532  df-gsum 17533  df-prds 17538  df-pws 17540  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-grp 19066  df-minusg 19067  df-sbg 19068  df-subg 19252  df-cntz 19450  df-cmn 19915  df-abl 19916  df-mgp 20280  df-rng 20294  df-ur 20327  df-ring 20380  df-subrg 20738  df-lmod 21052  df-lss 21122  df-sra 21363  df-rgmod 21364  df-dsmm 21951  df-frlm 21966  df-mnring 45058

This theorem is used by: (None)