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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnghmresel | Structured version Visualization version GIF version |
Description: An element of the non-unital ring homomorphisms restricted to a subset of non-unital rings is a non-unital ring homomorphisms. (Contributed by AV, 9-Mar-2020.) |
Ref | Expression |
---|---|
rnghmresel.h | ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rnghmresel | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHomo 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghmresel.h | . . . . . 6 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
2 | 1 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
3 | 2 | oveqd 6922 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌)) |
4 | ovres 7060 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHomo 𝑌)) | |
5 | 4 | adantl 475 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHomo 𝑌)) |
6 | 3, 5 | eqtrd 2861 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋 RngHomo 𝑌)) |
7 | 6 | eleq2d 2892 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌))) |
8 | 7 | biimp3a 1597 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHomo 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 × cxp 5340 ↾ cres 5344 (class class class)co 6905 RngHomo crngh 42725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-xp 5348 df-res 5354 df-iota 6086 df-fv 6131 df-ov 6908 |
This theorem is referenced by: rnghmsubcsetclem2 42816 |
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