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Theorem 1marepvmarrepid 20598
 Description: Replacing the ith row by 0's and the ith component of a (column) vector at the diagonal position for the identity matrix with the ith column replaced by the vector results in the matrix itself. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.)
Hypotheses
Ref Expression
marepvmarrep1.v 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
marepvmarrep1.o 1 = (1r‘(𝑁 Mat 𝑅))
marepvmarrep1.x 𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)
Assertion
Ref Expression
1marepvmarrepid (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍𝐼))𝐼) = 𝑋)

Proof of Theorem 1marepvmarrepid
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marepvmarrep1.x . . . 4 𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)
2 eqid 2770 . . . . . 6 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3 eqid 2770 . . . . . 6 (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
4 marepvmarrep1.v . . . . . 6 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁)
5 marepvmarrep1.o . . . . . 6 1 = (1r‘(𝑁 Mat 𝑅))
62, 3, 4, 5ma1repvcl 20593 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍𝑉𝐼𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
76ancom2s 621 . . . 4 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
81, 7syl5eqel 2853 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9 elmapi 8030 . . . . . . 7 (𝑍 ∈ ((Base‘𝑅) ↑𝑚 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10 ffvelrn 6500 . . . . . . . 8 ((𝑍:𝑁⟶(Base‘𝑅) ∧ 𝐼𝑁) → (𝑍𝐼) ∈ (Base‘𝑅))
1110ex 397 . . . . . . 7 (𝑍:𝑁⟶(Base‘𝑅) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
129, 11syl 17 . . . . . 6 (𝑍 ∈ ((Base‘𝑅) ↑𝑚 𝑁) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1312, 4eleq2s 2867 . . . . 5 (𝑍𝑉 → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1413impcom 394 . . . 4 ((𝐼𝑁𝑍𝑉) → (𝑍𝐼) ∈ (Base‘𝑅))
1514adantl 467 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑍𝐼) ∈ (Base‘𝑅))
16 simpl 468 . . . 4 ((𝐼𝑁𝑍𝑉) → 𝐼𝑁)
1716adantl 467 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝐼𝑁)
18 eqid 2770 . . . 4 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
19 eqid 2770 . . . 4 (0g𝑅) = (0g𝑅)
202, 3, 18, 19marrepval 20585 . . 3 (((𝑋 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝑍𝐼) ∈ (Base‘𝑅)) ∧ (𝐼𝑁𝐼𝑁)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍𝐼))𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗))))
218, 15, 17, 17, 20syl22anc 1476 . 2 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍𝐼))𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗))))
22 iftrue 4229 . . . . . 6 (𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)))
2322adantr 466 . . . . 5 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)))
24 iftrue 4229 . . . . . . . 8 (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (𝑍𝐼))
2524adantr 466 . . . . . . 7 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (𝑍𝐼))
26 iftrue 4229 . . . . . . . 8 (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝑖))
27 fveq2 6332 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑍𝑖) = (𝑍𝐼))
2827adantr 466 . . . . . . . 8 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑍𝑖) = (𝑍𝐼))
2926, 28sylan9eq 2824 . . . . . . 7 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝐼))
3025, 29eqtr4d 2807 . . . . . 6 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
31 eqid 2770 . . . . . . . . . . 11 (1r𝑅) = (1r𝑅)
32 simpr 471 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
3332adantr 466 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑁 ∈ Fin)
34333ad2ant1 1126 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
35 simpl 468 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
3635adantr 466 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑅 ∈ Ring)
37363ad2ant1 1126 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
38 simp2 1130 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
39 simp3 1131 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
402, 31, 19, 34, 37, 38, 39, 5mat1ov 20471 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
4140adantl 467 . . . . . . . . 9 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
4241adantl 467 . . . . . . . 8 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
43 eqtr2 2790 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝐼 = 𝑗)
4443eqcomd 2776 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝑗 = 𝐼)
4544ex 397 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖 = 𝑗𝑗 = 𝐼))
4645con3d 149 . . . . . . . . . . 11 (𝑖 = 𝐼 → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4746adantr 466 . . . . . . . . . 10 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4847impcom 394 . . . . . . . . 9 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → ¬ 𝑖 = 𝑗)
49 iffalse 4232 . . . . . . . . 9 𝑖 = 𝑗 → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (0g𝑅))
5048, 49syl 17 . . . . . . . 8 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (0g𝑅))
5142, 50eqtrd 2804 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → (𝑖 1 𝑗) = (0g𝑅))
52 iffalse 4232 . . . . . . . 8 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
5352adantr 466 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
54 iffalse 4232 . . . . . . . 8 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (0g𝑅))
5554adantr 466 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (0g𝑅))
5651, 53, 553eqtr4rd 2815 . . . . . 6 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
5730, 56pm2.61ian 795 . . . . 5 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
5823, 57eqtrd 2804 . . . 4 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
59 iffalse 4232 . . . . . 6 𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
6059adantr 466 . . . . 5 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
612, 3, 5mat1bas 20472 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
6261adantr 466 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
63 simpr 471 . . . . . . . . . . 11 ((𝐼𝑁𝑍𝑉) → 𝑍𝑉)
6463adantl 467 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑍𝑉)
6562, 64, 173jca 1121 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁))
66653ad2ant1 1126 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁))
67 3simpc 1145 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
6837, 66, 673jca 1121 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) ∧ (𝑖𝑁𝑗𝑁)))
6968adantl 467 . . . . . 6 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) ∧ (𝑖𝑁𝑗𝑁)))
702, 3, 4, 5, 19, 1ma1repveval 20594 . . . . . 6 ((𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖𝑋𝑗) = if(𝑗 = 𝐼, (𝑍𝑖), if(𝑗 = 𝑖, (1r𝑅), (0g𝑅))))
7169, 70syl 17 . . . . 5 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑖𝑋𝑗) = if(𝑗 = 𝐼, (𝑍𝑖), if(𝑗 = 𝑖, (1r𝑅), (0g𝑅))))
7234ad2antlr 698 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑁 ∈ Fin)
7337ad2antlr 698 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑅 ∈ Ring)
7438ad2antlr 698 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑖𝑁)
7539ad2antlr 698 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑗𝑁)
762, 31, 19, 72, 73, 74, 75, 5mat1ov 20471 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
77 equcom 2102 . . . . . . . . 9 (𝑖 = 𝑗𝑗 = 𝑖)
7877a1i 11 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 = 𝑗𝑗 = 𝑖))
7978ifbid 4245 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
8076, 79eqtr2d 2805 . . . . . 6 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)) = (𝑖 1 𝑗))
8180ifeq2da 4254 . . . . 5 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑗 = 𝐼, (𝑍𝑖), if(𝑗 = 𝑖, (1r𝑅), (0g𝑅))) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
8260, 71, 813eqtrd 2808 . . . 4 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
8358, 82pm2.61ian 795 . . 3 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
8483mpt2eq3dva 6865 . 2 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗))))
85 eqid 2770 . . . . 5 (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅)
862, 3, 85, 4marepvval 20590 . . . 4 (( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗))))
8765, 86syl 17 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗))))
881, 87syl5req 2817 . 2 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗))) = 𝑋)
8921, 84, 883eqtrd 2808 1 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍𝐼))𝐼) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  ifcif 4223  ⟶wf 6027  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794   ↑𝑚 cmap 8008  Fincfn 8108  Basecbs 16063  0gc0g 16307  1rcur 18708  Ringcrg 18754   Mat cmat 20429   matRRep cmarrep 20579   matRepV cmatrepV 20580 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-ot 4323  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-sup 8503  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-ip 16166  df-tset 16167  df-ple 16168  df-ds 16171  df-hom 16173  df-cco 16174  df-0g 16309  df-gsum 16310  df-prds 16315  df-pws 16317  df-mre 16453  df-mrc 16454  df-acs 16456  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-mhm 17542  df-submnd 17543  df-grp 17632  df-minusg 17633  df-sbg 17634  df-mulg 17748  df-subg 17798  df-ghm 17865  df-cntz 17956  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-subrg 18987  df-lmod 19074  df-lss 19142  df-sra 19386  df-rgmod 19387  df-dsmm 20292  df-frlm 20307  df-mamu 20406  df-mat 20430  df-marrep 20581  df-marepv 20582 This theorem is referenced by:  cramerimplem1  20707  cramerimplem1OLD  20708
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