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Theorem 1marepvmarrepid 20295
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 2626 . . . . . 6 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
3 eqid 2626 . . . . . 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 20290 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍𝑉𝐼𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
76ancom2s 843 . . . 4 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅)))
81, 7syl5eqel 2708 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅)))
9 elmapi 7824 . . . . . . 7 (𝑍 ∈ ((Base‘𝑅) ↑𝑚 𝑁) → 𝑍:𝑁⟶(Base‘𝑅))
10 ffvelrn 6314 . . . . . . . 8 ((𝑍:𝑁⟶(Base‘𝑅) ∧ 𝐼𝑁) → (𝑍𝐼) ∈ (Base‘𝑅))
1110ex 450 . . . . . . 7 (𝑍:𝑁⟶(Base‘𝑅) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
129, 11syl 17 . . . . . 6 (𝑍 ∈ ((Base‘𝑅) ↑𝑚 𝑁) → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1312, 4eleq2s 2722 . . . . 5 (𝑍𝑉 → (𝐼𝑁 → (𝑍𝐼) ∈ (Base‘𝑅)))
1413impcom 446 . . . 4 ((𝐼𝑁𝑍𝑉) → (𝑍𝐼) ∈ (Base‘𝑅))
1514adantl 482 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑍𝐼) ∈ (Base‘𝑅))
16 simpl 473 . . . 4 ((𝐼𝑁𝑍𝑉) → 𝐼𝑁)
1716adantl 482 . . 3 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝐼𝑁)
18 eqid 2626 . . . 4 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
19 eqid 2626 . . . 4 (0g𝑅) = (0g𝑅)
202, 3, 18, 19marrepval 20282 . . 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 1324 . 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 4069 . . . . . 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 4069 . . . . . . . 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 4069 . . . . . . . 8 (𝑗 = 𝐼 → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝑖))
27 fveq2 6150 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑍𝑖) = (𝑍𝐼))
2827adantr 481 . . . . . . . 8 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (𝑍𝑖) = (𝑍𝐼))
2926, 28sylan9eq 2680 . . . . . . 7 ((𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)) = (𝑍𝐼))
3025, 29eqtr4d 2663 . . . . . 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 2626 . . . . . . . . . . 11 (1r𝑅) = (1r𝑅)
32 simpr 477 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
3332adantr 481 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑁 ∈ Fin)
34333ad2ant1 1080 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
35 simpl 473 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑅 ∈ Ring)
3635adantr 481 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑅 ∈ Ring)
37363ad2ant1 1080 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
38 simp2 1060 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
39 simp3 1061 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
402, 31, 19, 34, 37, 38, 39, 5mat1ov 20168 . . . . . . . . . 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 2646 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝐼 = 𝑗)
4443eqcomd 2632 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑖 = 𝑗) → 𝑗 = 𝐼)
4544ex 450 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖 = 𝑗𝑗 = 𝐼))
4645con3d 148 . . . . . . . . . . 11 (𝑖 = 𝐼 → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4746adantr 481 . . . . . . . . . 10 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → (¬ 𝑗 = 𝐼 → ¬ 𝑖 = 𝑗))
4847impcom 446 . . . . . . . . 9 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → ¬ 𝑖 = 𝑗)
49 iffalse 4072 . . . . . . . . 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 2660 . . . . . . 7 ((¬ 𝑗 = 𝐼 ∧ (𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁))) → (𝑖 1 𝑗) = (0g𝑅))
52 iffalse 4072 . . . . . . . 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 4072 . . . . . . . 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 2671 . . . . . 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 830 . . . . 5 ((𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) → if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
5823, 57eqtrd 2660 . . . 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 4072 . . . . . 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 20169 . . . . . . . . . . 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 477 . . . . . . . . . . 11 ((𝐼𝑁𝑍𝑉) → 𝑍𝑉)
6463adantl 482 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → 𝑍𝑉)
6562, 64, 173jca 1240 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁))
66653ad2ant1 1080 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍𝑉𝐼𝑁))
67 3simpc 1058 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
6837, 66, 673jca 1240 . . . . . . 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 20291 . . . . . 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 762 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑁 ∈ Fin)
7337ad2antlr 762 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑅 ∈ Ring)
7438ad2antlr 762 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑖𝑁)
7539ad2antlr 762 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → 𝑗𝑁)
762, 31, 19, 72, 73, 74, 75, 5mat1ov 20168 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
77 equcom 1947 . . . . . . . . 9 (𝑖 = 𝑗𝑗 = 𝑖)
7877a1i 11 . . . . . . . 8 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → (𝑖 = 𝑗𝑗 = 𝑖))
7978ifbid 4085 . . . . . . 7 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)))
8076, 79eqtr2d 2661 . . . . . 6 (((¬ 𝑖 = 𝐼 ∧ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁)) ∧ ¬ 𝑗 = 𝐼) → if(𝑗 = 𝑖, (1r𝑅), (0g𝑅)) = (𝑖 1 𝑗))
8180ifeq2da 4094 . . . . 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 2664 . . . 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 830 . . 3 ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) ∧ 𝑖𝑁𝑗𝑁) → if(𝑖 = 𝐼, if(𝑗 = 𝐼, (𝑍𝐼), (0g𝑅)), (𝑖𝑋𝑗)) = if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗)))
8483mpt2eq3dva 6673 . 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 2626 . . . . 5 (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅)
862, 3, 85, 4marepvval 20287 . . . 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 2673 . 2 (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐼, (𝑍𝑖), (𝑖 1 𝑗))) = 𝑋)
8921, 84, 883eqtrd 2664 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 384  w3a 1036   = wceq 1480  wcel 1992  ifcif 4063  wf 5846  cfv 5850  (class class class)co 6605  cmpt2 6607  𝑚 cmap 7803  Fincfn 7900  Basecbs 15776  0gc0g 16016  1rcur 18417  Ringcrg 18463   Mat cmat 20127   matRRep cmarrep 20276   matRepV cmatrepV 20277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fsupp 8221  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-fzo 12404  df-seq 12739  df-hash 13055  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-sca 15873  df-vsca 15874  df-ip 15875  df-tset 15876  df-ple 15877  df-ds 15880  df-hom 15882  df-cco 15883  df-0g 16018  df-gsum 16019  df-prds 16024  df-pws 16026  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-sbg 17343  df-mulg 17457  df-subg 17507  df-ghm 17574  df-cntz 17666  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-ring 18465  df-subrg 18694  df-lmod 18781  df-lss 18847  df-sra 19086  df-rgmod 19087  df-dsmm 19990  df-frlm 20005  df-mamu 20104  df-mat 20128  df-marrep 20278  df-marepv 20279
This theorem is referenced by:  cramerimplem1  20403
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