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Theorem 1marepvmarrepid 20583
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 2760 . . . . . 6 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3 eqid 2760 . . . . . 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 20578 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍𝑉𝐼𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
76ancom2s 879 . . . 4 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
81, 7syl5eqel 2843 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9 elmapi 8045 . . . . . . 7 (𝑍 ∈ ((Base‘𝑅) ↑𝑚 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10 ffvelrn 6520 . . . . . . . 8 ((𝑍:𝑁⟶(Base‘𝑅) ∧ 𝐼𝑁) → (𝑍𝐼) ∈ (Base‘𝑅))
1110ex 449 . . . . . . 7 (𝑍:𝑁⟶(Base‘𝑅) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
129, 11syl 17 . . . . . 6 (𝑍 ∈ ((Base‘𝑅) ↑𝑚 𝑁) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1312, 4eleq2s 2857 . . . . 5 (𝑍𝑉 → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1413impcom 445 . . . 4 ((𝐼𝑁𝑍𝑉) → (𝑍𝐼) ∈ (Base‘𝑅))
1514adantl 473 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑍𝐼) ∈ (Base‘𝑅))
16 simpl 474 . . . 4 ((𝐼𝑁𝑍𝑉) → 𝐼𝑁)
1716adantl 473 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝐼𝑁)
18 eqid 2760 . . . 4 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
19 eqid 2760 . . . 4 (0g𝑅) = (0g𝑅)
202, 3, 18, 19marrepval 20570 . . 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 1478 . 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 4236 . . . . . 6 (𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)))
2322adantr 472 . . . . 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 4236 . . . . . . . 8 (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (𝑍𝐼))
2524adantr 472 . . . . . . 7 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (𝑍𝐼))
26 iftrue 4236 . . . . . . . 8 (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝑖))
27 fveq2 6352 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑍𝑖) = (𝑍𝐼))
2827adantr 472 . . . . . . . 8 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑍𝑖) = (𝑍𝐼))
2926, 28sylan9eq 2814 . . . . . . 7 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝐼))
3025, 29eqtr4d 2797 . . . . . 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 2760 . . . . . . . . . . 11 (1r𝑅) = (1r𝑅)
32 simpr 479 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
3332adantr 472 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑁 ∈ Fin)
34333ad2ant1 1128 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
35 simpl 474 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
3635adantr 472 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑅 ∈ Ring)
37363ad2ant1 1128 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
38 simp2 1132 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
39 simp3 1133 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
402, 31, 19, 34, 37, 38, 39, 5mat1ov 20456 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
4140adantl 473 . . . . . . . . 9 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
4241adantl 473 . . . . . . . 8 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
43 eqtr2 2780 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝐼 = 𝑗)
4443eqcomd 2766 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝑗 = 𝐼)
4544ex 449 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖 = 𝑗𝑗 = 𝐼))
4645con3d 148 . . . . . . . . . . 11 (𝑖 = 𝐼 → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4746adantr 472 . . . . . . . . . 10 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4847impcom 445 . . . . . . . . 9 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → ¬ 𝑖 = 𝑗)
49 iffalse 4239 . . . . . . . . 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 2794 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → (𝑖 1 𝑗) = (0g𝑅))
52 iffalse 4239 . . . . . . . 8 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
5352adantr 472 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗))
54 iffalse 4239 . . . . . . . 8 𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (0g𝑅))
5554adantr 472 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = (0g𝑅))
5651, 53, 553eqtr4rd 2805 . . . . . 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 866 . . . . 5 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
5823, 57eqtrd 2794 . . . 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 4239 . . . . . 6 𝑖 = 𝐼 → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
6059adantr 472 . . . . 5 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = (𝑖𝑋𝑗))
612, 3, 5mat1bas 20457 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
6261adantr 472 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 1 ∈ (Base‘(𝑁 Mat 𝑅)))
63 simpr 479 . . . . . . . . . . 11 ((𝐼𝑁𝑍𝑉) → 𝑍𝑉)
6463adantl 473 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑍𝑉)
6562, 64, 173jca 1123 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁))
66653ad2ant1 1128 . . . . . . . 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 1123 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) ∧ (𝑖𝑁𝑗𝑁)))
6968adantl 473 . . . . . 6 ((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑅 ∈ Ring ∧ ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁) ∧ (𝑖𝑁𝑗𝑁)))
702, 3, 4, 5, 19, 1ma1repveval 20579 . . . . . 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 765 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑁 ∈ Fin)
7337ad2antlr 765 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑅 ∈ Ring)
7438ad2antlr 765 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑖𝑁)
7539ad2antlr 765 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑗𝑁)
762, 31, 19, 72, 73, 74, 75, 5mat1ov 20456 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
77 equcom 2100 . . . . . . . . 9 (𝑖 = 𝑗𝑗 = 𝑖)
7877a1i 11 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 = 𝑗𝑗 = 𝑖))
7978ifbid 4252 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
8076, 79eqtr2d 2795 . . . . . 6 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)) = (𝑖 1 𝑗))
8180ifeq2da 4261 . . . . 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 2798 . . . 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 866 . . 3 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
8483mpt2eq3dva 6884 . 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 2760 . . . . 5 (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅)
862, 3, 85, 4marepvval 20575 . . . 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 2807 . 2 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗))) = 𝑋)
8921, 84, 883eqtrd 2798 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 383  w3a 1072   = wceq 1632  wcel 2139  ifcif 4230  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  Fincfn 8121  Basecbs 16059  0gc0g 16302  1rcur 18701  Ringcrg 18747   Mat cmat 20415   matRRep cmarrep 20564   matRepV cmatrepV 20565
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 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-hom 16168  df-cco 16169  df-0g 16304  df-gsum 16305  df-prds 16310  df-pws 16312  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17792  df-ghm 17859  df-cntz 17950  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-ring 18749  df-subrg 18980  df-lmod 19067  df-lss 19135  df-sra 19374  df-rgmod 19375  df-dsmm 20278  df-frlm 20293  df-mamu 20392  df-mat 20416  df-marrep 20566  df-marepv 20567
This theorem is referenced by:  cramerimplem1  20691
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