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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTVlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for rhmsubcALTV 48206. (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 5721 | . . . 4 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → 〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅)) | |
| 2 | 1 | 3adant1 1130 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → 〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅)) | 
| 3 | 2 | fvresd 6925 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) = ( RingHom ‘〈𝑋, 𝑌〉)) | 
| 4 | df-ov 7435 | . . 3 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
| 5 | rngcrescrhmALTV.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
| 6 | 5 | fveq1i 6906 | . . 3 ⊢ (𝐻‘〈𝑋, 𝑌〉) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) | 
| 7 | 4, 6 | eqtri 2764 | . 2 ⊢ (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) | 
| 8 | df-ov 7435 | . 2 ⊢ (𝑋 RingHom 𝑌) = ( RingHom ‘〈𝑋, 𝑌〉) | |
| 9 | 3, 7, 8 | 3eqtr4g 2801 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∩ cin 3949 〈cop 4631 × cxp 5682 ↾ cres 5686 ‘cfv 6560 (class class class)co 7432 Ringcrg 20231 RingHom crh 20470 RngCatALTVcrngcALTV 48184 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-res 5696 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: rhmsubcALTVlem3 48204 rhmsubcALTVlem4 48205 | 
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