Theorem marrepval 22910

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 22909	. . 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 486	. 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 783	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → 𝐾 ∈ 𝑁)
8		simplrr 790	. . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ 𝑘 = 𝐾) → 𝐿 ∈ 𝑁)
9	1, 2	matrcl 22760	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
10	9	simpld 500	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
11	10, 10	jca 521	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
12	11	ad3antrrr 743	. . . 4 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
13		mpoexga 8118	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))) ∈ V)
14	12, 13	syl 18	. . 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 2775	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑖 = 𝑘 ↔ 𝑖 = 𝐾))
16	15	adantr 486	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → (𝑖 = 𝑘 ↔ 𝑖 = 𝐾))
17		eqeq2 2775	. . . . . . . 8 ⊢ (𝑙 = 𝐿 → (𝑗 = 𝑙 ↔ 𝑗 = 𝐿))
18	17	ifbid 4509	. . . . . . 7 ⊢ (𝑙 = 𝐿 → if(𝑗 = 𝑙, 𝑆, 0 ) = if(𝑗 = 𝐿, 𝑆, 0 ))
19	18	adantl 487	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → if(𝑗 = 𝑙, 𝑆, 0 ) = if(𝑗 = 𝐿, 𝑆, 0 ))
20	16, 19	ifbieq1d 4510	. . . . 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 7525	. . . 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 487	. . 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 7605	. 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 16	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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ifcif 4485  ‘cfv 6561  (class class class)co 7446   ∈ cmpo 7448  Fincfn 9004  Basecbs 17419  0_gc0g 17642   Mat cmat 22755   matRRep cmarrep 22904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-1cn 11288  ax-addcl 11290

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-nn 12366  df-slot 17392  df-ndx 17404  df-base 17420  df-mat 22756  df-marrep 22906

This theorem is used by:  marrepeval  22911  marrepcl  22912  1marepvmarrepid  22923  smadiadetglem1  23019  smadiadetglem2  23020  madjusmdetlem1  34542