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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 7278 | . 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 21628 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {〈𝑂, 𝑋〉}) |
8 | 7 | fveq1d 6776 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘〈𝐸, 𝐸〉) = ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉)) |
9 | 5 | eqcomi 2747 | . . . . 5 ⊢ 〈𝐸, 𝐸〉 = 𝑂 |
10 | 9 | fveq2i 6777 | . . . 4 ⊢ ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉) = ({〈𝑂, 𝑋〉}‘𝑂) |
11 | opex 5379 | . . . . . 6 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
12 | 5, 11 | eqeltri 2835 | . . . . 5 ⊢ 𝑂 ∈ V |
13 | simp3 1137 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
14 | fvsng 7052 | . . . . 5 ⊢ ((𝑂 ∈ V ∧ 𝑋 ∈ 𝐾) → ({〈𝑂, 𝑋〉}‘𝑂) = 𝑋) | |
15 | 12, 13, 14 | sylancr 587 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({〈𝑂, 𝑋〉}‘𝑂) = 𝑋) |
16 | 10, 15 | eqtrid 2790 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉) = 𝑋) |
17 | 8, 16 | eqtrd 2778 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘〈𝐸, 𝐸〉) = 𝑋) |
18 | 1, 17 | eqtrid 2790 | 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 1539 ∈ wcel 2106 Vcvv 3432 {csn 4561 〈cop 4567 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Ringcrg 19783 Mat cmat 21554 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 |
This theorem is referenced by: mat1ghm 21632 mat1mhm 21633 |
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