Theorem marrepval 21711

Description: Third substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)

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
marrepval	⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))

Distinct variable groups:   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝑖,𝑀,𝑗   𝑆,𝑖,𝑗   𝑖,𝐾,𝑗   𝑖,𝐿,𝑗

Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐵(𝑖,𝑗)   𝑄(𝑖,𝑗)   0 (𝑖,𝑗)

Proof of Theorem marrepval

Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marrepfval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		marrepfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
3		marrepfval.q	. . . 4 ⊢ 𝑄 = (𝑁 matRRep 𝑅)
4		marrepfval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
5	1, 2, 3, 4	marrepval0 21710	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
6	5	adantr 481	. 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
7		simprl 768	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → 𝐾 ∈ 𝑁)
8		simplrr 775	. . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ 𝑘 = 𝐾) → 𝐿 ∈ 𝑁)
9	1, 2	matrcl 21559	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
10	9	simpld 495	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
11	10, 10	jca 512	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
12	11	ad3antrrr 727	. . . 4 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
13		mpoexga 7918	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) ∈ V)
14	12, 13	syl 17	. . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) ∈ V)
15		eqeq2 2750	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑖 = 𝑘 ↔ 𝑖 = 𝐾))
16	15	adantr 481	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → (𝑖 = 𝑘 ↔ 𝑖 = 𝐾))
17		eqeq2 2750	. . . . . . . 8 ⊢ (𝑙 = 𝐿 → (𝑗 = 𝑙 ↔ 𝑗 = 𝐿))
18	17	ifbid 4482	. . . . . . 7 ⊢ (𝑙 = 𝐿 → if(𝑗 = 𝑙, 𝑆, 0 ) = if(𝑗 = 𝐿, 𝑆, 0 ))
19	18	adantl 482	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → if(𝑗 = 𝑙, 𝑆, 0 ) = if(𝑗 = 𝐿, 𝑆, 0 ))
20	16, 19	ifbieq1d 4483	. . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)) = if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)))
21	20	mpoeq3dv 7354	. . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
22	21	adantl 482	. . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
23	7, 8, 14, 22	ovmpodv2 7431	. 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → ((𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)))))
24	6, 23	mpd 15	1 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ifcif 4459  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Fincfn 8733  Basecbs 16912  0_gc0g 17150   Mat cmat 21554   matRRep cmarrep 21705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-1cn 10929  ax-addcl 10931

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-nn 11974  df-slot 16883  df-ndx 16895  df-base 16913  df-mat 21555  df-marrep 21707

This theorem is referenced by:  marrepeval  21712  marrepcl  21713  1marepvmarrepid  21724  smadiadetglem1  21820  smadiadetglem2  21821  madjusmdetlem1  31777