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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 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 22485 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {〈𝑂, 𝑋〉}) | 
| 8 | 7 | fveq1d 6908 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘〈𝐸, 𝐸〉) = ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉)) | 
| 9 | 5 | eqcomi 2746 | . . . . 5 ⊢ 〈𝐸, 𝐸〉 = 𝑂 | 
| 10 | 9 | fveq2i 6909 | . . . 4 ⊢ ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉) = ({〈𝑂, 𝑋〉}‘𝑂) | 
| 11 | opex 5469 | . . . . . 6 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
| 12 | 5, 11 | eqeltri 2837 | . . . . 5 ⊢ 𝑂 ∈ V | 
| 13 | simp3 1139 | . . . . 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 2789 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ({〈𝑂, 𝑋〉}‘〈𝐸, 𝐸〉) = 𝑋) | 
| 17 | 8, 16 | eqtrd 2777 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → ((𝐹‘𝑋)‘〈𝐸, 𝐸〉) = 𝑋) | 
| 18 | 1, 17 | eqtrid 2789 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Ringcrg 20230 Mat cmat 22411 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: mat1ghm 22489 mat1mhm 22490 | 
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