Theorem marepvfval 22530

Description: First substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

Ref	Expression
marepvfval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
marepvfval.b	⊢ 𝐵 = (Base‘𝐴)
marepvfval.q	⊢ 𝑄 = (𝑁 matRepV 𝑅)
marepvfval.v	⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)

Assertion

Ref	Expression
marepvfval	⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))

Distinct variable groups:   𝐵,𝑚,𝑣   𝑖,𝑁,𝑗,𝑘,𝑚,𝑣   𝑅,𝑖,𝑗,𝑘,𝑚,𝑣   𝑚,𝑉,𝑣

Allowed substitution hints:   𝐴(𝑣,𝑖,𝑗,𝑘,𝑚)   𝐵(𝑖,𝑗,𝑘)   𝑄(𝑣,𝑖,𝑗,𝑘,𝑚)   𝑉(𝑖,𝑗,𝑘)

Proof of Theorem marepvfval

Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marepvfval.q	. 2 ⊢ 𝑄 = (𝑁 matRepV 𝑅)
2		marepvfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
3	2	fvexi 6854	. . . . 5 ⊢ 𝐵 ∈ V
4		marepvfval.v	. . . . . . 7 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)
5	4	ovexi 7401	. . . . . 6 ⊢ 𝑉 ∈ V
6	5	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝑉 ∈ V)
7		mpoexga 8030	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑉 ∈ V) → (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) ∈ V)
8	3, 6, 7	sylancr 588	. . . 4 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) ∈ V)
9		oveq12 7376	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
10		marepvfval.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
11	9, 10	eqtr4di 2789	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
12	11	fveq2d 6844	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
13	12, 2	eqtr4di 2789	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
14		fveq2 6840	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
15	14	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = (Base‘𝑅))
16		simpl 482	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
17	15, 16	oveq12d 7385	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑅) ↑m 𝑁))
18	17, 4	eqtr4di 2789	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((Base‘𝑟) ↑m 𝑛) = 𝑉)
19		eqidd 2737	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))
20	16, 16, 19	mpoeq123dv 7442	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))
21	16, 20	mpteq12dv 5172	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))
22	13, 18, 21	mpoeq123dv 7442	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))))
23		df-marepv 22524	. . . . 5 ⊢ matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))))
24	22, 23	ovmpoga 7521	. . . 4 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V ∧ (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) ∈ V) → (𝑁 matRepV 𝑅) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))))
25	8, 24	mpd3an3 1465	. . 3 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))))
26	23	mpondm0 7607	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = ∅)
27	10	fveq2i 6843	. . . . . . . 8 ⊢ (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
28	2, 27	eqtri 2759	. . . . . . 7 ⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅))
29		matbas0pc 22374	. . . . . . 7 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
30	28, 29	eqtrid 2783	. . . . . 6 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
31	30	orcd 874	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝐵 = ∅ ∨ 𝑉 = ∅))
32		0mpo0 7450	. . . . 5 ⊢ ((𝐵 = ∅ ∨ 𝑉 = ∅) → (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) = ∅)
33	31, 32	syl 17	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) = ∅)
34	26, 33	eqtr4d 2774	. . 3 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))))
35	25, 34	pm2.61i 182	. 2 ⊢ (𝑁 matRepV 𝑅) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))
36	1, 35	eqtri 2759	1 ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ∅c0 4273  ifcif 4466   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369   ↑_m cmap 8773  Basecbs 17179   Mat cmat 22372   matRepV cmatrepV 22522

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-1cn 11096  ax-addcl 11098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-nn 12175  df-slot 17152  df-ndx 17164  df-base 17180  df-mat 22373  df-marepv 22524

This theorem is referenced by:  marepvval0  22531