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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTVlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for rhmsubcALTV 48129. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngcrescrhmALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcrescrhmALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) |
rngcrescrhmALTV.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) |
rngcrescrhmALTV.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) |
Ref | Expression |
---|---|
rhmsubcALTVlem2 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5726 | . . . 4 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → 〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅)) | |
2 | 1 | 3adant1 1129 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → 〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅)) |
3 | 2 | fvresd 6927 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) = ( RingHom ‘〈𝑋, 𝑌〉)) |
4 | df-ov 7434 | . . 3 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
5 | rngcrescrhmALTV.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
6 | 5 | fveq1i 6908 | . . 3 ⊢ (𝐻‘〈𝑋, 𝑌〉) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) |
7 | 4, 6 | eqtri 2763 | . 2 ⊢ (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) |
8 | df-ov 7434 | . 2 ⊢ (𝑋 RingHom 𝑌) = ( RingHom ‘〈𝑋, 𝑌〉) | |
9 | 3, 7, 8 | 3eqtr4g 2800 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 〈cop 4637 × cxp 5687 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 Ringcrg 20251 RingHom crh 20486 RngCatALTVcrngcALTV 48107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-res 5701 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: rhmsubcALTVlem3 48127 rhmsubcALTVlem4 48128 |
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