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Theorem 1marepvmarrepid 22068
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 2732 . . . . . 6 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3 eqid 2732 . . . . . 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 22063 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍𝑉𝐼𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
76ancom2s 648 . . . 4 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
81, 7eqeltrid 2837 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9 elmapi 8839 . . . . . . 7 (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10 ffvelcdm 7080 . . . . . . . 8 ((𝑍:𝑁⟶(Base‘𝑅) ∧ 𝐼𝑁) → (𝑍𝐼) ∈ (Base‘𝑅))
1110ex 413 . . . . . . 7 (𝑍:𝑁⟶(Base‘𝑅) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
129, 11syl 17 . . . . . 6 (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1312, 4eleq2s 2851 . . . . 5 (𝑍𝑉 → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1413impcom 408 . . . 4 ((𝐼𝑁𝑍𝑉) → (𝑍𝐼) ∈ (Base‘𝑅))
1514adantl 482 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑍𝐼) ∈ (Base‘𝑅))
16 simpl 483 . . . 4 ((𝐼𝑁𝑍𝑉) → 𝐼𝑁)
1716adantl 482 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝐼𝑁)
18 eqid 2732 . . . 4 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
19 eqid 2732 . . . 4 (0g𝑅) = (0g𝑅)
202, 3, 18, 19marrepval 22055 . . 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 837 . 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 4533 . . . . . 6 (𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)))
2322adantr 481 . . . . 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 4533 . . . . . . . 8 (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (𝑍𝐼))
2524adantr 481 . . . . . . 7 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (𝑍𝐼))
26 iftrue 4533 . . . . . . . 8 (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝑖))
27 fveq2 6888 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑍𝑖) = (𝑍𝐼))
2827adantr 481 . . . . . . . 8 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑍𝑖) = (𝑍𝐼))
2926, 28sylan9eq 2792 . . . . . . 7 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝐼))
3025, 29eqtr4d 2775 . . . . . 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 2732 . . . . . . . . . . 11 (1r𝑅) = (1r𝑅)
32 simpr 485 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
3332adantr 481 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑁 ∈ Fin)
34333ad2ant1 1133 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
35 simpl 483 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
3635adantr 481 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑅 ∈ Ring)
37363ad2ant1 1133 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
38 simp2 1137 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
39 simp3 1138 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
402, 31, 19, 34, 37, 38, 39, 5mat1ov 21941 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
4140adantl 482 . . . . . . . . 9 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
4241adantl 482 . . . . . . . 8 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
43 eqtr2 2756 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝐼 = 𝑗)
4443eqcomd 2738 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝑗 = 𝐼)
4544ex 413 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖 = 𝑗𝑗 = 𝐼))
4645con3d 152 . . . . . . . . . . 11 (𝑖 = 𝐼 → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4746adantr 481 . . . . . . . . . 10 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4847impcom 408 . . . . . . . . 9 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → ¬ 𝑖 = 𝑗)
49 iffalse 4536 . . . . . . . . 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 2772 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → (𝑖 1 𝑗) = (0g𝑅))
52 iffalse 4536 . . . . . . . 8 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
5352adantr 481 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
54 iffalse 4536 . . . . . . . 8 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (0g𝑅))
5554adantr 481 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (0g𝑅))
5651, 53, 553eqtr4rd 2783 . . . . . 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 810 . . . . 5 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
5823, 57eqtrd 2772 . . . 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 4536 . . . . . 6 𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
6059adantr 481 . . . . 5 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
612, 3, 5mat1bas 21942 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
6261adantr 481 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
63 simpr 485 . . . . . . . . . . 11 ((𝐼𝑁𝑍𝑉) → 𝑍𝑉)
6463adantl 482 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑍𝑉)
6562, 64, 173jca 1128 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁))
66653ad2ant1 1133 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁))
67 3simpc 1150 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
6837, 66, 673jca 1128 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) ∧ (𝑖𝑁𝑗𝑁)))
6968adantl 482 . . . . . 6 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) ∧ (𝑖𝑁𝑗𝑁)))
702, 3, 4, 5, 19, 1ma1repveval 22064 . . . . . 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 725 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑁 ∈ Fin)
7337ad2antlr 725 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑅 ∈ Ring)
7438ad2antlr 725 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑖𝑁)
7539ad2antlr 725 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑗𝑁)
762, 31, 19, 72, 73, 74, 75, 5mat1ov 21941 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
77 equcom 2021 . . . . . . . . 9 (𝑖 = 𝑗𝑗 = 𝑖)
7877a1i 11 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 = 𝑗𝑗 = 𝑖))
7978ifbid 4550 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
8076, 79eqtr2d 2773 . . . . . 6 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)) = (𝑖 1 𝑗))
8180ifeq2da 4559 . . . . 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 2776 . . . 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 810 . . 3 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
8483mpoeq3dva 7482 . 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 2732 . . . . 5 (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅)
862, 3, 85, 4marepvval 22060 . . . 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, 87eqtr2id 2785 . 2 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗))) = 𝑋)
8921, 84, 883eqtrd 2776 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  ifcif 4527  wf 6536  cfv 6540  (class class class)co 7405  cmpo 7407  m cmap 8816  Fincfn 8935  Basecbs 17140  0gc0g 17381  1rcur 19998  Ringcrg 20049   Mat cmat 21898   matRRep cmarrep 22049   matRepV cmatrepV 22050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-hom 17217  df-cco 17218  df-0g 17383  df-gsum 17384  df-prds 17389  df-pws 17391  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-subrg 20353  df-lmod 20465  df-lss 20535  df-sra 20777  df-rgmod 20778  df-dsmm 21278  df-frlm 21293  df-mamu 21877  df-mat 21899  df-marrep 22051  df-marepv 22052
This theorem is referenced by:  cramerimplem1  22176
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