Theorem mamufv 22736

Description: A cell in the 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)))
mamufv.i	⊢ (𝜑 → 𝐼 ∈ 𝑀)
mamufv.k	⊢ (𝜑 → 𝐾 ∈ 𝑃)

Assertion

Ref	Expression
mamufv	⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))

Distinct variable groups:   𝑗,𝑀   𝑗,𝑁   𝑃,𝑗   𝑅,𝑗   𝑗,𝑋   𝑗,𝑌   𝜑,𝑗   𝑗,𝐼   𝑗,𝐾

Allowed substitution hints:   𝐵(𝑗)   · (𝑗)   𝐹(𝑗)   𝑉(𝑗)

Proof of Theorem mamufv

Dummy variables 𝑖 𝑘 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		mamuval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁)))
9		mamuval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mamuval 22735	. 2 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
11		oveq1 7447	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
12		oveq2 7448	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑗𝑌𝑘) = (𝑗𝑌𝐾))
13	11, 12	oveqan12d 7459	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑘 = 𝐾) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))
14	13	adantl 487	. . . 4 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))
15	14	mpteq2dv 5202	. . 3 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))
16	15	oveq2d 7456	. 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))
17		mamufv.i	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑀)
18		mamufv.k	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝑃)
19		ovexd 7475	. 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))) ∈ V)
20	10, 16, 17, 18, 19	ovmpod 7593	1 ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cotp 4861   ↦ cmpt 5189   × cxp 5653  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872  Fincfn 8998  Basecbs 17413  ._rcmulr 17455   Σ_g cgsu 17637   maMul cmmul 22732

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-mamu 22733

This theorem is used by:  mamuass  22744  mamudi  22745  mamudir  22746  mamuvs1  22747  mamuvs2  22748  mamulid  22783  mamurid  22784  matmulcell  22787  mavmulass  22891  mvmumamul1  22896  mdetmul  22965  matunitlindflem2  23022  decpmatmullem  23116