Theorem marrepval0 20590

Description: Second 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
marrepval0	⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))

Distinct variable groups:   𝑖,𝑁,𝑗,𝑘,𝑙   𝑅,𝑖,𝑗,𝑘,𝑙   𝑖,𝑀,𝑗,𝑘,𝑙   𝑆,𝑖,𝑗,𝑘,𝑙

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

Proof of Theorem marrepval0

Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marrepfval.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		marrepfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
3	1, 2	matrcl 20441	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 477	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
5	4, 4	jca 555	. . . 4 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
6	5	adantr 472	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
7		mpt2exga 7416	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V)
8	6, 7	syl 17	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V)
9		ifeq1 4235	. . . . . . 7 ⊢ (𝑠 = 𝑆 → if(𝑗 = 𝑙, 𝑠, 0 ) = if(𝑗 = 𝑙, 𝑆, 0 ))
10	9	adantl 473	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → if(𝑗 = 𝑙, 𝑠, 0 ) = if(𝑗 = 𝑙, 𝑆, 0 ))
11		oveq 6821	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
12	11	adantr 472	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
13	10, 12	ifeq12d 4251	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))
14	13	mpt2eq3dv 6888	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → (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))))
15	14	mpt2eq3dv 6888	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
16		marrepfval.q	. . . 4 ⊢ 𝑄 = (𝑁 matRRep 𝑅)
17		marrepfval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
18	1, 2, 16, 17	marrepfval 20589	. . 3 ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
19	15, 18	ovmpt2ga 6957	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
20	8, 19	mpd3an3 1574	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2140  Vcvv 3341  ifcif 4231  ‘cfv 6050  (class class class)co 6815   ↦ cmpt2 6817  Fincfn 8124  Basecbs 16080  0_gc0g 16323   Mat cmat 20436   matRRep cmarrep 20585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-slot 16084  df-base 16086  df-mat 20437  df-marrep 20587

This theorem is referenced by:  marrepval  20591  minmar1marrep  20679