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Theorem 1marepvmarrepid 22568
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 2726 . . . . . 6 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3 eqid 2726 . . . . . 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 22563 . . . . 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 2830 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9 elmapi 8878 . . . . . . 7 (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10 ffvelcdm 7095 . . . . . . . 8 ((𝑍:𝑁⟶(Base‘𝑅) ∧ 𝐼𝑁) → (𝑍𝐼) ∈ (Base‘𝑅))
1110ex 411 . . . . . . 7 (𝑍:𝑁⟶(Base‘𝑅) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
129, 11syl 17 . . . . . 6 (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1312, 4eleq2s 2844 . . . . 5 (𝑍𝑉 → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1413impcom 406 . . . 4 ((𝐼𝑁𝑍𝑉) → (𝑍𝐼) ∈ (Base‘𝑅))
1514adantl 480 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑍𝐼) ∈ (Base‘𝑅))
16 simpl 481 . . . 4 ((𝐼𝑁𝑍𝑉) → 𝐼𝑁)
1716adantl 480 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝐼𝑁)
18 eqid 2726 . . . 4 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
19 eqid 2726 . . . 4 (0g𝑅) = (0g𝑅)
202, 3, 18, 19marrepval 22555 . . 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 4539 . . . . . 6 (𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)))
2322adantr 479 . . . . 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 4539 . . . . . . . 8 (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (𝑍𝐼))
2524adantr 479 . . . . . . 7 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (𝑍𝐼))
26 iftrue 4539 . . . . . . . 8 (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝑖))
27 fveq2 6901 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑍𝑖) = (𝑍𝐼))
2827adantr 479 . . . . . . . 8 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑍𝑖) = (𝑍𝐼))
2926, 28sylan9eq 2786 . . . . . . 7 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝐼))
3025, 29eqtr4d 2769 . . . . . 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 2726 . . . . . . . . . . 11 (1r𝑅) = (1r𝑅)
32 simpr 483 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
3332adantr 479 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑁 ∈ Fin)
34333ad2ant1 1130 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
35 simpl 481 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
3635adantr 479 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑅 ∈ Ring)
37363ad2ant1 1130 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
38 simp2 1134 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
39 simp3 1135 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
402, 31, 19, 34, 37, 38, 39, 5mat1ov 22441 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
4140adantl 480 . . . . . . . . 9 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
4241adantl 480 . . . . . . . 8 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
43 eqtr2 2750 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝐼 = 𝑗)
4443eqcomd 2732 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝑗 = 𝐼)
4544ex 411 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖 = 𝑗𝑗 = 𝐼))
4645con3d 152 . . . . . . . . . . 11 (𝑖 = 𝐼 → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4746adantr 479 . . . . . . . . . 10 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4847impcom 406 . . . . . . . . 9 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → ¬ 𝑖 = 𝑗)
49 iffalse 4542 . . . . . . . . 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 2766 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → (𝑖 1 𝑗) = (0g𝑅))
52 iffalse 4542 . . . . . . . 8 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
5352adantr 479 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
54 iffalse 4542 . . . . . . . 8 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (0g𝑅))
5554adantr 479 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (0g𝑅))
5651, 53, 553eqtr4rd 2777 . . . . . 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 2766 . . . 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 4542 . . . . . 6 𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
6059adantr 479 . . . . 5 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
612, 3, 5mat1bas 22442 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
6261adantr 479 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
63 simpr 483 . . . . . . . . . . 11 ((𝐼𝑁𝑍𝑉) → 𝑍𝑉)
6463adantl 480 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑍𝑉)
6562, 64, 173jca 1125 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁))
66653ad2ant1 1130 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁))
67 3simpc 1147 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
6837, 66, 673jca 1125 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) ∧ (𝑖𝑁𝑗𝑁)))
6968adantl 480 . . . . . 6 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) ∧ (𝑖𝑁𝑗𝑁)))
702, 3, 4, 5, 19, 1ma1repveval 22564 . . . . . 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 22441 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
77 equcom 2014 . . . . . . . . 9 (𝑖 = 𝑗𝑗 = 𝑖)
7877a1i 11 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 = 𝑗𝑗 = 𝑖))
7978ifbid 4556 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
8076, 79eqtr2d 2767 . . . . . 6 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)) = (𝑖 1 𝑗))
8180ifeq2da 4565 . . . . 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 2770 . . . 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 7502 . 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 2726 . . . . 5 (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅)
862, 3, 85, 4marepvval 22560 . . . 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 2779 . 2 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗))) = 𝑋)
8921, 84, 883eqtrd 2770 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 394  w3a 1084   = wceq 1534  wcel 2099  ifcif 4533  wf 6550  cfv 6554  (class class class)co 7424  cmpo 7426  m cmap 8855  Fincfn 8974  Basecbs 17213  0gc0g 17454  1rcur 20164  Ringcrg 20216   Mat cmat 22398   matRRep cmarrep 22549   matRepV cmatrepV 22550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-ot 4642  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-sup 9485  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-sca 17282  df-vsca 17283  df-ip 17284  df-tset 17285  df-ple 17286  df-ds 17288  df-hom 17290  df-cco 17291  df-0g 17456  df-gsum 17457  df-prds 17462  df-pws 17464  df-mre 17599  df-mrc 17600  df-acs 17602  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-mhm 18773  df-submnd 18774  df-grp 18931  df-minusg 18932  df-sbg 18933  df-mulg 19062  df-subg 19117  df-ghm 19207  df-cntz 19311  df-cmn 19780  df-abl 19781  df-mgp 20118  df-rng 20136  df-ur 20165  df-ring 20218  df-subrg 20553  df-lmod 20838  df-lss 20909  df-sra 21151  df-rgmod 21152  df-dsmm 21730  df-frlm 21745  df-mamu 22382  df-mat 22399  df-marrep 22551  df-marepv 22552
This theorem is referenced by:  cramerimplem1  22676
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