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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 7406 | . 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 21962 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {〈𝑂, 𝑋〉}) |
8 | 7 | fveq1d 6889 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘〈𝐸, 𝐸〉) = ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉)) |
9 | 5 | eqcomi 2742 | . . . . 5 ⊢ 〈𝐸, 𝐸〉 = 𝑂 |
10 | 9 | fveq2i 6890 | . . . 4 ⊢ ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉) = ({〈𝑂, 𝑋〉}‘𝑂) |
11 | opex 5462 | . . . . . 6 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
12 | 5, 11 | eqeltri 2830 | . . . . 5 ⊢ 𝑂 ∈ V |
13 | simp3 1139 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
14 | fvsng 7172 | . . . . 5 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐾) → ({〈𝑂, 𝑋〉}‘𝑂) = 𝑋) | |
15 | 12, 13, 14 | sylancr 588 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({〈𝑂, 𝑋〉}‘𝑂) = 𝑋) |
16 | 10, 15 | eqtrid 2785 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉) = 𝑋) |
17 | 8, 16 | eqtrd 2773 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘〈𝐸, 𝐸〉) = 𝑋) |
18 | 1, 17 | eqtrid 2785 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4626 〈cop 4632 ↦ cmpt 5229 ‘cfv 6539 (class class class)co 7403 Basecbs 17139 Ringcrg 20046 Mat cmat 21888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6491 df-fun 6541 df-fv 6547 df-ov 7406 |
This theorem is referenced by: mat1ghm 21966 mat1mhm 21967 |
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