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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 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
3 | 2 | oveqd 7394 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌)) |
4 | ovres 7540 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHomo 𝑌)) | |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHomo 𝑌)) |
6 | 3, 5 | eqtrd 2771 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋 RngHomo 𝑌)) |
7 | 6 | eleq2d 2818 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌))) |
8 | 7 | biimp3a 1469 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHomo 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 × cxp 5651 ↾ cres 5655 (class class class)co 7377 RngHomo crngh 46336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pr 5404 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-br 5126 df-opab 5188 df-xp 5659 df-res 5665 df-iota 6468 df-fv 6524 df-ov 7380 |
This theorem is referenced by: rnghmsubcsetclem2 46427 |
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