Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mat1rhmval | 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 |
---|---|
mat1rhmval | ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {〈𝑂, 𝑋〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mat1rhmval.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉})) |
3 | opeq2 4341 | . . . 4 ⊢ (𝑥 = 𝑋 → 〈𝑂, 𝑥〉 = 〈𝑂, 𝑋〉) | |
4 | 3 | sneqd 4137 | . . 3 ⊢ (𝑥 = 𝑋 → {〈𝑂, 𝑥〉} = {〈𝑂, 𝑋〉}) |
5 | 4 | adantl 481 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) ∧ 𝑥 = 𝑋) → {〈𝑂, 𝑥〉} = {〈𝑂, 𝑋〉}) |
6 | simp3 1056 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾) | |
7 | snex 4835 | . . 3 ⊢ {〈𝑂, 𝑋〉} ∈ V | |
8 | 7 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → {〈𝑂, 𝑋〉} ∈ V) |
9 | 2, 5, 6, 8 | fvmptd 6197 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {〈𝑂, 𝑋〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Ringcrg 18370 Mat cmat 20032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: mat1rhmelval 20105 mat1rhmcl 20106 mat1mhm 20109 |
Copyright terms: Public domain | W3C validator |