![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7434 | . 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 22501 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {〈𝑂, 𝑋〉}) |
8 | 7 | fveq1d 6909 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘〈𝐸, 𝐸〉) = ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉)) |
9 | 5 | eqcomi 2744 | . . . . 5 ⊢ 〈𝐸, 𝐸〉 = 𝑂 |
10 | 9 | fveq2i 6910 | . . . 4 ⊢ ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉) = ({〈𝑂, 𝑋〉}‘𝑂) |
11 | opex 5475 | . . . . . 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 7200 | . . . . 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 2787 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉) = 𝑋) |
17 | 8, 16 | eqtrd 2775 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘〈𝐸, 𝐸〉) = 𝑋) |
18 | 1, 17 | eqtrid 2787 | 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 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Ringcrg 20251 Mat cmat 22427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 |
This theorem is referenced by: mat1ghm 22505 mat1mhm 22506 |
Copyright terms: Public domain | W3C validator |