Theorem marepvfval 22481

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 6842	. . . . 5 ⊢ 𝐵 ∈ V
4		marepvfval.v	. . . . . . 7 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)
5	4	ovexi 7386	. . . . . 6 ⊢ 𝑉 ∈ V
6	5	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝑉 ∈ V)
7		mpoexga 8015	. . . . 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 587	. . . 4 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) ∈ V)
9		oveq12 7361	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
10		marepvfval.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
11	9, 10	eqtr4di 2786	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
12	11	fveq2d 6832	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
13	12, 2	eqtr4di 2786	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
14		fveq2 6828	. . . . . . . . 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 7370	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑅) ↑m 𝑁))
18	17, 4	eqtr4di 2786	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((Base‘𝑟) ↑m 𝑛) = 𝑉)
19		eqidd 2734	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))
20	16, 16, 19	mpoeq123dv 7427	. . . . . . 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 5180	. . . . . 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 7427	. . . . 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 22475	. . . . 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 7506	. . . 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 1464	. . 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 7592	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = ∅)
27	10	fveq2i 6831	. . . . . . . 8 ⊢ (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
28	2, 27	eqtri 2756	. . . . . . 7 ⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅))
29		matbas0pc 22325	. . . . . . 7 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
30	28, 29	eqtrid 2780	. . . . . 6 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
31	30	orcd 873	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝐵 = ∅ ∨ 𝑉 = ∅))
32		0mpo0 7435	. . . . 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 2771	. . 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 2756	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 847   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282  ifcif 4474   ↦ cmpt 5174  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354   ↑_m cmap 8756  Basecbs 17122   Mat cmat 22323   matRepV cmatrepV 22473

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  ax-cnex 11069  ax-1cn 11071  ax-addcl 11073

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  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-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  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-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-nn 12133  df-slot 17095  df-ndx 17107  df-base 17123  df-mat 22324  df-marepv 22475

This theorem is referenced by:  marepvval0  22482