Theorem mamufv 22279

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 22278	. 2 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
11		oveq1 7356	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
12		oveq2 7357	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑗𝑌𝑘) = (𝑗𝑌𝐾))
13	11, 12	oveqan12d 7368	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑘 = 𝐾) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))
14	13	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)) = ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))
15	14	mpteq2dv 5186	. . 3 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))
16	15	oveq2d 7365	. 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 7384	. 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))) ∈ V)
20	10, 16, 17, 18, 19	ovmpod 7501	1 ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ⟨cotp 4585   ↦ cmpt 5173   × cxp 5617  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  Fincfn 8872  Basecbs 17120  ._rcmulr 17162   Σ_g cgsu 17344   maMul cmmul 22275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-mamu 22276

This theorem is referenced by:  mamuass  22287  mamudi  22288  mamudir  22289  mamuvs1  22290  mamuvs2  22291  mamulid  22326  mamurid  22327  matmulcell  22330  mavmulass  22434  mvmumamul1  22439  mdetmul  22508  decpmatmullem  22656  matunitlindflem2  37617