Theorem mamuval 22688

Description: Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)

Hypotheses

Ref	Expression
mamufval.f	⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)
mamufval.b	⊢ 𝐵 = (Base‘𝑅)
mamufval.t	⊢ · = (.r‘𝑅)
mamufval.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
mamufval.m	⊢ (𝜑 → 𝑀 ∈ Fin)
mamufval.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mamufval.p	⊢ (𝜑 → 𝑃 ∈ Fin)
mamuval.x	⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))
mamuval.y	⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)))

Assertion

Ref	Expression
mamuval	⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))

Distinct variable groups:   𝑖,𝑗,𝑘,𝑀   𝑖,𝑁,𝑗,𝑘   𝑃,𝑖,𝑗,𝑘   𝑅,𝑖,𝑗,𝑘   𝑖,𝑋,𝑗,𝑘   𝑖,𝑌,𝑗,𝑘   𝜑,𝑖,𝑗,𝑘   · ,𝑖,𝑘

Allowed substitution hints:   𝐵(𝑖, 𝑗, 𝑘)   · (𝑗)   𝐹(𝑖, 𝑗, 𝑘)   𝑉(𝑖, 𝑗, 𝑘)

Proof of Theorem mamuval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mamufval.f	. . 3 ⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)
2		mamufval.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		mamufval.t	. . 3 ⊢ · = (.r‘𝑅)
4		mamufval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
5		mamufval.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ Fin)
6		mamufval.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
7		mamufval.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ Fin)
8	1, 2, 3, 4, 5, 6, 7	mamufval 22687	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m (𝑁 × 𝑃)) ↦ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
9		oveq 7418	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗))
10		oveq 7418	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑗𝑦𝑘) = (𝑗𝑌𝑘))
11	9, 10	oveqan12d 7431	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)) = ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))
12	11	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)) = ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))
13	12	mpteq2dv 5199	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))
14	13	oveq2d 7428	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))))
15	14	mpoeq3dv 7491	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))))) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
16		mamuval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))
17		mamuval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)))
18		mpoexga 8079	. . 3 ⊢ ((𝑀 ∈ Fin ∧ 𝑃 ∈ Fin) → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))) ∈ V)
19	5, 7, 18	syl2anc 596	. 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))) ∈ V)
20	8, 15, 16, 17, 19	ovmpod 7564	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cotp 4592   ↦ cmpt 5186   × cxp 5649  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8831  Fincfn 8957  Basecbs 17367  ._rcmulr 17409   Σ_g cgsu 17591   maMul cmmul 22685

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-mamu 22686

This theorem is used by:  mamufv  22689  mamures  22692  mamucl  22696  mpomatmul  22741  mamutpos  22753  mat1dimmul  22771  dmatmul  22792  madurid  22939  cramerimplem2  22982  mat2pmatmul  23029  decpmatmul  23070