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Mirrors > Home > MPE Home > Th. List > mat1rhmelval | Structured version Visualization version GIF version |
Description: The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.) |
Ref | Expression |
---|---|
mat1rhmval.k | ⊢ 𝐾 = (Base‘𝑅) |
mat1rhmval.a | ⊢ 𝐴 = ({𝐸} Mat 𝑅) |
mat1rhmval.b | ⊢ 𝐵 = (Base‘𝐴) |
mat1rhmval.o | ⊢ 𝑂 = ⟨𝐸, 𝐸⟩ |
mat1rhmval.f | ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {⟨𝑂, 𝑥⟩}) |
Ref | Expression |
---|---|
mat1rhmelval | ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7415 | . 2 ⊢ (𝐸(𝐹‘𝑋)𝐸) = ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) | |
2 | mat1rhmval.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
3 | mat1rhmval.a | . . . . 5 ⊢ 𝐴 = ({𝐸} Mat 𝑅) | |
4 | mat1rhmval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
5 | mat1rhmval.o | . . . . 5 ⊢ 𝑂 = ⟨𝐸, 𝐸⟩ | |
6 | mat1rhmval.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {⟨𝑂, 𝑥⟩}) | |
7 | 2, 3, 4, 5, 6 | mat1rhmval 22202 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {⟨𝑂, 𝑋⟩}) |
8 | 7 | fveq1d 6894 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩)) |
9 | 5 | eqcomi 2740 | . . . . 5 ⊢ ⟨𝐸, 𝐸⟩ = 𝑂 |
10 | 9 | fveq2i 6895 | . . . 4 ⊢ ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = ({⟨𝑂, 𝑋⟩}‘𝑂) |
11 | opex 5465 | . . . . . 6 ⊢ ⟨𝐸, 𝐸⟩ ∈ V | |
12 | 5, 11 | eqeltri 2828 | . . . . 5 ⊢ 𝑂 ∈ V |
13 | simp3 1137 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
14 | fvsng 7181 | . . . . 5 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋) | |
15 | 12, 13, 14 | sylancr 586 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘𝑂) = 𝑋) |
16 | 10, 15 | eqtrid 2783 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({⟨𝑂, 𝑋⟩}‘⟨𝐸, 𝐸⟩) = 𝑋) |
17 | 8, 16 | eqtrd 2771 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘⟨𝐸, 𝐸⟩) = 𝑋) |
18 | 1, 17 | eqtrid 2783 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 Vcvv 3473 {csn 4629 ⟨cop 4635 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7412 Basecbs 17149 Ringcrg 20128 Mat cmat 22128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7415 |
This theorem is referenced by: mat1ghm 22206 mat1mhm 22207 |
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