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Theorem mvmulfval 21891
Description: Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.)
Hypotheses
Ref Expression
mvmulfval.x × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
mvmulfval.b 𝐵 = (Base‘𝑅)
mvmulfval.t · = (.r𝑅)
mvmulfval.r (𝜑𝑅𝑉)
mvmulfval.m (𝜑𝑀 ∈ Fin)
mvmulfval.n (𝜑𝑁 ∈ Fin)
Assertion
Ref Expression
mvmulfval (𝜑× = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
Distinct variable groups:   𝑖,𝑗,𝑥,𝑦,𝜑   𝑖,𝑀,𝑗,𝑥,𝑦   𝑖,𝑁,𝑗,𝑥,𝑦   𝑅,𝑖,𝑗,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥, · ,𝑦,𝑖
Allowed substitution hints:   𝐵(𝑖,𝑗)   · (𝑗)   × (𝑥,𝑦,𝑖,𝑗)   𝑉(𝑥,𝑦,𝑖,𝑗)

Proof of Theorem mvmulfval
Dummy variables 𝑚 𝑛 𝑜 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvmulfval.x . 2 × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
2 df-mvmul 21890 . . . 4 maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))))
32a1i 11 . . 3 (𝜑 → maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))))))
4 fvex 6855 . . . . 5 (1st𝑜) ∈ V
5 fvex 6855 . . . . 5 (2nd𝑜) ∈ V
6 xpeq12 5658 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑚 × 𝑛) = ((1st𝑜) × (2nd𝑜)))
76oveq2d 7373 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → ((Base‘𝑟) ↑m (𝑚 × 𝑛)) = ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))))
8 oveq2 7365 . . . . . . 7 (𝑛 = (2nd𝑜) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑟) ↑m (2nd𝑜)))
98adantl 482 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑟) ↑m (2nd𝑜)))
10 simpl 483 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → 𝑚 = (1st𝑜))
11 simpr 485 . . . . . . . . 9 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → 𝑛 = (2nd𝑜))
1211mpteq1d 5200 . . . . . . . 8 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))) = (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))
1312oveq2d 7373 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))) = (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))
1410, 13mpteq12dv 5196 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))) = (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))))
157, 9, 14mpoeq123dv 7432 . . . . 5 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑m (2nd𝑜)) ↦ (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))))
164, 5, 15csbie2 3897 . . . 4 (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑m (2nd𝑜)) ↦ (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))))
17 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → 𝑟 = 𝑅)
1817fveq2d 6846 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = (Base‘𝑅))
19 mvmulfval.b . . . . . . 7 𝐵 = (Base‘𝑅)
2018, 19eqtr4di 2794 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = 𝐵)
21 fveq2 6842 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁⟩ → (1st𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
2221ad2antll 727 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
23 mvmulfval.m . . . . . . . . . 10 (𝜑𝑀 ∈ Fin)
24 mvmulfval.n . . . . . . . . . 10 (𝜑𝑁 ∈ Fin)
25 op1stg 7933 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2623, 24, 25syl2anc 584 . . . . . . . . 9 (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2726adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2822, 27eqtrd 2776 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) = 𝑀)
29 fveq2 6842 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁⟩ → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
3029ad2antll 727 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
31 op2ndg 7934 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3223, 24, 31syl2anc 584 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3332adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3430, 33eqtrd 2776 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd𝑜) = 𝑁)
3528, 34xpeq12d 5664 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((1st𝑜) × (2nd𝑜)) = (𝑀 × 𝑁))
3620, 35oveq12d 7375 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))) = (𝐵m (𝑀 × 𝑁)))
3720, 34oveq12d 7375 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑m (2nd𝑜)) = (𝐵m 𝑁))
38 fveq2 6842 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
3938adantr 481 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩) → (.r𝑟) = (.r𝑅))
4039adantl 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (.r𝑟) = (.r𝑅))
41 mvmulfval.t . . . . . . . . . 10 · = (.r𝑅)
4240, 41eqtr4di 2794 . . . . . . . . 9 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (.r𝑟) = · )
4342oveqd 7374 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)) = ((𝑖𝑥𝑗) · (𝑦𝑗)))
4434, 43mpteq12dv 5196 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))) = (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))
4517, 44oveq12d 7375 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))
4628, 45mpteq12dv 5196 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))))
4736, 37, 46mpoeq123dv 7432 . . . 4 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑥 ∈ ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑m (2nd𝑜)) ↦ (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))) = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
4816, 47eqtrid 2788 . . 3 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))) = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
49 mvmulfval.r . . . 4 (𝜑𝑅𝑉)
5049elexd 3465 . . 3 (𝜑𝑅 ∈ V)
51 opex 5421 . . . 4 𝑀, 𝑁⟩ ∈ V
5251a1i 11 . . 3 (𝜑 → ⟨𝑀, 𝑁⟩ ∈ V)
53 ovex 7390 . . . . 5 (𝐵m (𝑀 × 𝑁)) ∈ V
54 ovex 7390 . . . . 5 (𝐵m 𝑁) ∈ V
5553, 54mpoex 8012 . . . 4 (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))) ∈ V
5655a1i 11 . . 3 (𝜑 → (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))) ∈ V)
573, 48, 50, 52, 56ovmpod 7507 . 2 (𝜑 → (𝑅 maVecMul ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
581, 57eqtrid 2788 1 (𝜑× = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  csb 3855  cop 4592  cmpt 5188   × cxp 5631  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  m cmap 8765  Fincfn 8883  Basecbs 17083  .rcmulr 17134   Σg cgsu 17322   maVecMul cmvmul 21889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-mvmul 21890
This theorem is referenced by:  mvmulval  21892  mavmuldm  21899  mavmul0g  21902
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