Theorem marrepfval 22582

Description: First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

Ref	Expression
marrepfval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
marrepfval.b	⊢ 𝐵 = (Base‘𝐴)
marrepfval.q	⊢ 𝑄 = (𝑁 matRRep 𝑅)
marrepfval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
marrepfval	⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))

Distinct variable groups:   𝐵,𝑚,𝑠   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚,𝑠   𝑅,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠

Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑙)   𝑄(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   0 (𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)

Proof of Theorem marrepfval

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

Step	Hyp	Ref	Expression
1		marrepfval.q	. 2 ⊢ 𝑄 = (𝑁 matRRep 𝑅)
2		marrepfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
3	2	fvexi 6921	. . . . 5 ⊢ 𝐵 ∈ V
4		fvexd 6922	. . . . 5 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
5		mpoexga 8101	. . . . 5 ⊢ ((𝐵 ∈ V ∧ (Base‘𝑅) ∈ V) → (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) ∈ V)
6	3, 4, 5	sylancr 587	. . . 4 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) ∈ V)
7		oveq12 7440	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
8	7	fveq2d 6911	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
9		marrepfval.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
10	9	fveq2i 6910	. . . . . . . 8 ⊢ (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
11	2, 10	eqtri 2763	. . . . . . 7 ⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅))
12	8, 11	eqtr4di 2793	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
13		fveq2 6907	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
14	13	adantl 481	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = (Base‘𝑅))
15		simpl 482	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
16		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
17		marrepfval.z	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝑅)
18	16, 17	eqtr4di 2793	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
19	18	ifeq2d 4551	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)) = if(𝑗 = 𝑙, 𝑠, 0 ))
20	19	ifeq1d 4550	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))
21	20	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))
22	15, 15, 21	mpoeq123dv 7508	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)), (𝑖𝑚𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))
23	15, 15, 22	mpoeq123dv 7508	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)), (𝑖𝑚𝑗)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
24	12, 14, 23	mpoeq123dv 7508	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)), (𝑖𝑚𝑗))))) = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))))
25		df-marrep 22580	. . . . 5 ⊢ matRRep = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)), (𝑖𝑚𝑗))))))
26	24, 25	ovmpoga 7587	. . . 4 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V ∧ (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) ∈ V) → (𝑁 matRRep 𝑅) = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))))
27	6, 26	mpd3an3 1461	. . 3 ⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRRep 𝑅) = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))))
28	25	mpondm0 7673	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRRep 𝑅) = ∅)
29		matbas0pc 22429	. . . . . . 7 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
30	11, 29	eqtrid 2787	. . . . . 6 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
31	30	orcd 873	. . . . 5 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝐵 = ∅ ∨ (Base‘𝑅) = ∅))
32		0mpo0 7516	. . . . 5 ⊢ ((𝐵 = ∅ ∨ (Base‘𝑅) = ∅) → (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) = ∅)
33	31, 32	syl 17	. . . 4 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) = ∅)
34	28, 33	eqtr4d 2778	. . 3 ⊢ (¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRRep 𝑅) = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))))
35	27, 34	pm2.61i 182	. 2 ⊢ (𝑁 matRRep 𝑅) = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
36	1, 35	eqtri 2763	1 ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  ifcif 4531  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17245  0_gc0g 17486   Mat cmat 22427   matRRep cmarrep 22578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-mat 22428  df-marrep 22580

This theorem is referenced by:  marrepval0  22583