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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmresel | Structured version Visualization version GIF version |
Description: An element of the unital ring homomorphisms restricted to a subset of unital rings is a unital ring homomorphism. (Contributed by AV, 10-Mar-2020.) |
Ref | Expression |
---|---|
rhmresel.h | ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rhmresel | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RingHom 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmresel.h | . . . . . 6 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
2 | 1 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
3 | 2 | oveqd 6991 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌)) |
4 | ovres 7128 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌)) | |
5 | 4 | adantl 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋( RingHom ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RingHom 𝑌)) |
6 | 3, 5 | eqtrd 2808 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) |
7 | 6 | eleq2d 2845 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋 RingHom 𝑌))) |
8 | 7 | biimp3a 1448 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RingHom 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 × cxp 5401 ↾ cres 5405 (class class class)co 6974 RingHom crh 19199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-xp 5409 df-res 5415 df-iota 6149 df-fv 6193 df-ov 6977 |
This theorem is referenced by: rhmsubcsetclem2 43682 rhmsubcrngclem2 43688 |
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