Theorem mamufv 20012

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 20011	. 2 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
11		oveq1 6556	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
12		oveq2 6557	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑗𝑌𝑘) = (𝑗𝑌𝐾))
13	11, 12	oveqan12d 6568	. . . . 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 4673	. . 3 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑘 = 𝐾)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾))))
16	15	oveq2d 6565	. 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		ovex 6577	. . 3 ⊢ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))) ∈ V
20	19	a1i 11	. 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))) ∈ V)
21	10, 16, 17, 18, 20	ovmpt2d 6686	1 ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cotp 4133   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  Fincfn 7841  Basecbs 15695  ._rcmulr 15769   Σ_g cgsu 15924   maMul cmmul 20008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-mamu 20009

This theorem is referenced by:  mamuass  20027  mamudi  20028  mamudir  20029  mamuvs1  20030  mamuvs2  20031  mamulid  20066  mamurid  20067  matmulcell  20070  mavmulass  20174  mvmumamul1  20179  mdetmul  20248  decpmatmullem  20395  matunitlindflem2  32576