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Theorem mamuvs1 22260

Description: Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)

**Hypotheses**
Ref	Expression
mamucl.b	⊢ 𝐵 = (Base‘𝑅)
mamucl.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mamudi.f	⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)
mamudi.m	⊢ (𝜑 → 𝑀 ∈ Fin)
mamudi.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mamudi.o	⊢ (𝜑 → 𝑂 ∈ Fin)
mamuvs1.t	⊢ · = (._r‘𝑅)
mamuvs1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mamuvs1.y	⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑_m (𝑀 × 𝑁)))
mamuvs1.z	⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑_m (𝑁 × 𝑂)))

**Assertion**
Ref	Expression
mamuvs1	⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)))

Proof of Theorem mamuvs1 Dummy variables 𝑖 𝑘 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mamucl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
2		eqid 2726	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
3		mamuvs1.t	. . . . . 6 ⊢ · = (._r‘𝑅)
4		mamucl.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑅 ∈ Ring)
6		mamudi.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Fin)
7	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑁 ∈ Fin)
8		mamuvs1.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑋 ∈ 𝐵)
10	4	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
11		mamuvs1.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑_m (𝑀 × 𝑁)))
12		elmapi 8845	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑_m (𝑀 × 𝑁)) → 𝑌:(𝑀 × 𝑁)⟶𝐵)
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌:(𝑀 × 𝑁)⟶𝐵)
14	13	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑌:(𝑀 × 𝑁)⟶𝐵)
15		simplrl 774	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑀)
16		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
17	14, 15, 16	fovcdmd 7576	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑌𝑗) ∈ 𝐵)
18		mamuvs1.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑_m (𝑁 × 𝑂)))
19		elmapi 8845	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝐵 ↑_m (𝑁 × 𝑂)) → 𝑍:(𝑁 × 𝑂)⟶𝐵)
20	18, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍:(𝑁 × 𝑂)⟶𝐵)
21	20	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑍:(𝑁 × 𝑂)⟶𝐵)
22		simplrr 775	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑘 ∈ 𝑂)
23	21, 16, 22	fovcdmd 7576	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑗𝑍𝑘) ∈ 𝐵)
24	1, 3, 10, 17, 23	ringcld 20162	. . . . . 6 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)) ∈ 𝐵)
25		eqid 2726	. . . . . . 7 ⊢ (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)))
26		ovexd 7440	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)) ∈ V)
27		fvexd 6900	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (0_g‘𝑅) ∈ V)
28	25, 7, 26, 27	fsuppmptdm 9376	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))) finSupp (0_g‘𝑅))
29	1, 2, 3, 5, 7, 9, 24, 28	gsummulc2 20216	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))) = (𝑋 · (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))))
30		df-ov 7408	. . . . . . . . . 10 ⊢ (𝑖(((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝑗) = ((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)‘⟨𝑖, 𝑗⟩)
31		simprl 768	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑖 ∈ 𝑀)
32		opelxpi 5706	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝑀 ∧ 𝑗 ∈ 𝑁) → ⟨𝑖, 𝑗⟩ ∈ (𝑀 × 𝑁))
33	31, 32	sylan 579	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ⟨𝑖, 𝑗⟩ ∈ (𝑀 × 𝑁))
34		mamudi.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ Fin)
35		xpfi 9319	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑀 × 𝑁) ∈ Fin)
36	34, 6, 35	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 × 𝑁) ∈ Fin)
37	36	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑀 × 𝑁) ∈ Fin)
38	8	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐵)
39		ffn 6711	. . . . . . . . . . . . . 14 ⊢ (𝑌:(𝑀 × 𝑁)⟶𝐵 → 𝑌 Fn (𝑀 × 𝑁))
40	11, 12, 39	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 Fn (𝑀 × 𝑁))
41	40	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → 𝑌 Fn (𝑀 × 𝑁))
42		df-ov 7408	. . . . . . . . . . . . . 14 ⊢ (𝑖𝑌𝑗) = (𝑌‘⟨𝑖, 𝑗⟩)
43	42	eqcomi 2735	. . . . . . . . . . . . 13 ⊢ (𝑌‘⟨𝑖, 𝑗⟩) = (𝑖𝑌𝑗)
44	43	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) ∧ ⟨𝑖, 𝑗⟩ ∈ (𝑀 × 𝑁)) → (𝑌‘⟨𝑖, 𝑗⟩) = (𝑖𝑌𝑗))
45	37, 38, 41, 44	ofc1 7693	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) ∧ ⟨𝑖, 𝑗⟩ ∈ (𝑀 × 𝑁)) → ((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)‘⟨𝑖, 𝑗⟩) = (𝑋 · (𝑖𝑌𝑗)))
46	33, 45	mpdan 684	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)‘⟨𝑖, 𝑗⟩) = (𝑋 · (𝑖𝑌𝑗)))
47	30, 46	eqtrid 2778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → (𝑖(((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝑗) = (𝑋 · (𝑖𝑌𝑗)))
48	47	oveq1d 7420	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝑗) · (𝑗𝑍𝑘)) = ((𝑋 · (𝑖𝑌𝑗)) · (𝑗𝑍𝑘)))
49	1, 3	ringass 20158	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ (𝑖𝑌𝑗) ∈ 𝐵 ∧ (𝑗𝑍𝑘) ∈ 𝐵)) → ((𝑋 · (𝑖𝑌𝑗)) · (𝑗𝑍𝑘)) = (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))
50	10, 38, 17, 23, 49	syl13anc 1369	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑋 · (𝑖𝑌𝑗)) · (𝑗𝑍𝑘)) = (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))
51	48, 50	eqtrd 2766	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ 𝑗 ∈ 𝑁) → ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝑗) · (𝑗𝑍𝑘)) = (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))
52	51	mpteq2dva 5241	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑗 ∈ 𝑁 ↦ ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝑗) · (𝑗𝑍𝑘))) = (𝑗 ∈ 𝑁 ↦ (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)))))
53	52	oveq2d 7421	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝑗) · (𝑗𝑍𝑘)))) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ (𝑋 · ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))))
54		mamudi.f	. . . . . . 7 ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)
55	34	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑀 ∈ Fin)
56		mamudi.o	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ Fin)
57	56	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑂 ∈ Fin)
58	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑌 ∈ (𝐵 ↑_m (𝑀 × 𝑁)))
59	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑍 ∈ (𝐵 ↑_m (𝑁 × 𝑂)))
60		simprr 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → 𝑘 ∈ 𝑂)
61	54, 1, 3, 5, 55, 7, 57, 58, 59, 31, 60	mamufv 22244	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖(𝑌𝐹𝑍)𝑘) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘)))))
62	61	oveq2d 7421	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘)) = (𝑋 · (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑌𝑗) · (𝑗𝑍𝑘))))))
63	29, 53, 62	3eqtr4d 2776	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝑗) · (𝑗𝑍𝑘)))) = (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘)))
64		fconst6g 6774	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐵 → ((𝑀 × 𝑁) × {𝑋}):(𝑀 × 𝑁)⟶𝐵)
65	8, 64	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑀 × 𝑁) × {𝑋}):(𝑀 × 𝑁)⟶𝐵)
66	1	fvexi 6899	. . . . . . . . 9 ⊢ 𝐵 ∈ V
67		elmapg 8835	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ (𝑀 × 𝑁) ∈ Fin) → (((𝑀 × 𝑁) × {𝑋}) ∈ (𝐵 ↑_m (𝑀 × 𝑁)) ↔ ((𝑀 × 𝑁) × {𝑋}):(𝑀 × 𝑁)⟶𝐵))
68	66, 36, 67	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → (((𝑀 × 𝑁) × {𝑋}) ∈ (𝐵 ↑_m (𝑀 × 𝑁)) ↔ ((𝑀 × 𝑁) × {𝑋}):(𝑀 × 𝑁)⟶𝐵))
69	65, 68	mpbird 257	. . . . . . 7 ⊢ (𝜑 → ((𝑀 × 𝑁) × {𝑋}) ∈ (𝐵 ↑_m (𝑀 × 𝑁)))
70	1, 3	ringvcl 22255	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((𝑀 × 𝑁) × {𝑋}) ∈ (𝐵 ↑_m (𝑀 × 𝑁)) ∧ 𝑌 ∈ (𝐵 ↑_m (𝑀 × 𝑁))) → (((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌) ∈ (𝐵 ↑_m (𝑀 × 𝑁)))
71	4, 69, 11, 70	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌) ∈ (𝐵 ↑_m (𝑀 × 𝑁)))
72	71	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌) ∈ (𝐵 ↑_m (𝑀 × 𝑁)))
73	54, 1, 3, 5, 55, 7, 57, 72, 59, 31, 60	mamufv 22244	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍)𝑘) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖(((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝑗) · (𝑗𝑍𝑘)))))
74		df-ov 7408	. . . . 5 ⊢ (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍))𝑘) = ((((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍))‘⟨𝑖, 𝑘⟩)
75		opelxpi 5706	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂) → ⟨𝑖, 𝑘⟩ ∈ (𝑀 × 𝑂))
76	75	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → ⟨𝑖, 𝑘⟩ ∈ (𝑀 × 𝑂))
77		xpfi 9319	. . . . . . . . 9 ⊢ ((𝑀 ∈ Fin ∧ 𝑂 ∈ Fin) → (𝑀 × 𝑂) ∈ Fin)
78	34, 56, 77	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝑀 × 𝑂) ∈ Fin)
79	78	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑀 × 𝑂) ∈ Fin)
80	1, 4, 54, 34, 6, 56, 11, 18	mamucl 22256	. . . . . . . . 9 ⊢ (𝜑 → (𝑌𝐹𝑍) ∈ (𝐵 ↑_m (𝑀 × 𝑂)))
81		elmapi 8845	. . . . . . . . 9 ⊢ ((𝑌𝐹𝑍) ∈ (𝐵 ↑_m (𝑀 × 𝑂)) → (𝑌𝐹𝑍):(𝑀 × 𝑂)⟶𝐵)
82		ffn 6711	. . . . . . . . 9 ⊢ ((𝑌𝐹𝑍):(𝑀 × 𝑂)⟶𝐵 → (𝑌𝐹𝑍) Fn (𝑀 × 𝑂))
83	80, 81, 82	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑌𝐹𝑍) Fn (𝑀 × 𝑂))
84	83	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑌𝐹𝑍) Fn (𝑀 × 𝑂))
85		df-ov 7408	. . . . . . . . 9 ⊢ (𝑖(𝑌𝐹𝑍)𝑘) = ((𝑌𝐹𝑍)‘⟨𝑖, 𝑘⟩)
86	85	eqcomi 2735	. . . . . . . 8 ⊢ ((𝑌𝐹𝑍)‘⟨𝑖, 𝑘⟩) = (𝑖(𝑌𝐹𝑍)𝑘)
87	86	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ ⟨𝑖, 𝑘⟩ ∈ (𝑀 × 𝑂)) → ((𝑌𝐹𝑍)‘⟨𝑖, 𝑘⟩) = (𝑖(𝑌𝐹𝑍)𝑘))
88	79, 9, 84, 87	ofc1 7693	. . . . . 6 ⊢ (((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) ∧ ⟨𝑖, 𝑘⟩ ∈ (𝑀 × 𝑂)) → ((((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍))‘⟨𝑖, 𝑘⟩) = (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘)))
89	76, 88	mpdan 684	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → ((((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍))‘⟨𝑖, 𝑘⟩) = (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘)))
90	74, 89	eqtrid 2778	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍))𝑘) = (𝑋 · (𝑖(𝑌𝐹𝑍)𝑘)))
91	63, 73, 90	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑀 ∧ 𝑘 ∈ 𝑂)) → (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍)𝑘) = (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍))𝑘))
92	91	ralrimivva 3194	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑂 (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍)𝑘) = (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍))𝑘))
93	1, 4, 54, 34, 6, 56, 71, 18	mamucl 22256	. . . 4 ⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍) ∈ (𝐵 ↑_m (𝑀 × 𝑂)))
94		elmapi 8845	. . . 4 ⊢ (((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍) ∈ (𝐵 ↑_m (𝑀 × 𝑂)) → ((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍):(𝑀 × 𝑂)⟶𝐵)
95		ffn 6711	. . . 4 ⊢ (((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍):(𝑀 × 𝑂)⟶𝐵 → ((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍) Fn (𝑀 × 𝑂))
96	93, 94, 95	3syl 18	. . 3 ⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍) Fn (𝑀 × 𝑂))
97		fconst6g 6774	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → ((𝑀 × 𝑂) × {𝑋}):(𝑀 × 𝑂)⟶𝐵)
98	8, 97	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑀 × 𝑂) × {𝑋}):(𝑀 × 𝑂)⟶𝐵)
99		elmapg 8835	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ (𝑀 × 𝑂) ∈ Fin) → (((𝑀 × 𝑂) × {𝑋}) ∈ (𝐵 ↑_m (𝑀 × 𝑂)) ↔ ((𝑀 × 𝑂) × {𝑋}):(𝑀 × 𝑂)⟶𝐵))
100	66, 78, 99	sylancr 586	. . . . . 6 ⊢ (𝜑 → (((𝑀 × 𝑂) × {𝑋}) ∈ (𝐵 ↑_m (𝑀 × 𝑂)) ↔ ((𝑀 × 𝑂) × {𝑋}):(𝑀 × 𝑂)⟶𝐵))
101	98, 100	mpbird 257	. . . . 5 ⊢ (𝜑 → ((𝑀 × 𝑂) × {𝑋}) ∈ (𝐵 ↑_m (𝑀 × 𝑂)))
102	1, 3	ringvcl 22255	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((𝑀 × 𝑂) × {𝑋}) ∈ (𝐵 ↑_m (𝑀 × 𝑂)) ∧ (𝑌𝐹𝑍) ∈ (𝐵 ↑_m (𝑀 × 𝑂))) → (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)) ∈ (𝐵 ↑_m (𝑀 × 𝑂)))
103	4, 101, 80, 102	syl3anc 1368	. . . 4 ⊢ (𝜑 → (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)) ∈ (𝐵 ↑_m (𝑀 × 𝑂)))
104		elmapi 8845	. . . 4 ⊢ ((((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)) ∈ (𝐵 ↑_m (𝑀 × 𝑂)) → (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)):(𝑀 × 𝑂)⟶𝐵)
105		ffn 6711	. . . 4 ⊢ ((((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)):(𝑀 × 𝑂)⟶𝐵 → (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)) Fn (𝑀 × 𝑂))
106	103, 104, 105	3syl 18	. . 3 ⊢ (𝜑 → (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)) Fn (𝑀 × 𝑂))
107		eqfnov2 7535	. . 3 ⊢ ((((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍) Fn (𝑀 × 𝑂) ∧ (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)) Fn (𝑀 × 𝑂)) → (((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)) ↔ ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑂 (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍)𝑘) = (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍))𝑘)))
108	96, 106, 107	syl2anc 583	. 2 ⊢ (𝜑 → (((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)) ↔ ∀𝑖 ∈ 𝑀 ∀𝑘 ∈ 𝑂 (𝑖((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍)𝑘) = (𝑖(((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍))𝑘)))
109	92, 108	mpbird 257	1 ⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘_f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘_f · (𝑌𝐹𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 {csn 4623 ⟨cop 4629 ⟨cotp 4631 ↦ cmpt 5224 × cxp 5667 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ∘_f cof 7665 ↑_m cmap 8822 Fincfn 8941 Basecbs 17153 ._rcmulr 17207 0_gc0g 17394 Σ_g cgsu 17395 Ringcrg 20138 maMul cmmul 22240

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 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 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-0g 17396 df-gsum 17397 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-minusg 18867 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-ur 20087 df-ring 20140 df-mamu 22241

This theorem is referenced by: matassa 22301 mdetmul 22480

W3C validator