Theorem mamuval 22309

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 22308	. 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 7358	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗))
10		oveq 7358	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑗𝑦𝑘) = (𝑗𝑌𝑘))
11	9, 10	oveqan12d 7371	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)) = ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))
12	11	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)) = ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))
13	12	mpteq2dv 5187	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))
14	13	oveq2d 7368	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘)))))
15	14	mpoeq3dv 7431	. 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 8015	. . 3 ⊢ ((𝑀 ∈ Fin ∧ 𝑃 ∈ Fin) → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))) ∈ V)
19	5, 7, 18	syl2anc 584	. 2 ⊢ (𝜑 → (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))) ∈ V)
20	8, 15, 16, 17, 19	ovmpod 7504	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ⟨cotp 4583   ↦ cmpt 5174   × cxp 5617  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354   ↑_m cmap 8756  Fincfn 8875  Basecbs 17122  ._rcmulr 17164   Σ_g cgsu 17346   maMul cmmul 22306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-ot 4584  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  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 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-mamu 22307

This theorem is referenced by:  mamufv  22310  mamures  22313  mamucl  22317  mpomatmul  22362  mamutpos  22374  mat1dimmul  22392  dmatmul  22413  madurid  22560  cramerimplem2  22600  mat2pmatmul  22647  decpmatmul  22688