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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTVlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for rhmsubcALTV 41901. (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 5072 | . . . 4 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → 〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅)) | |
2 | 1 | 3adant1 1072 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → 〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅)) |
3 | fvres 6117 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ (𝑅 × 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) = ( RingHom ‘〈𝑋, 𝑌〉)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) = ( RingHom ‘〈𝑋, 𝑌〉)) |
5 | df-ov 6552 | . . 3 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
6 | rngcrescrhmALTV.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
7 | 6 | fveq1i 6104 | . . 3 ⊢ (𝐻‘〈𝑋, 𝑌〉) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) |
8 | 5, 7 | eqtri 2632 | . 2 ⊢ (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) |
9 | df-ov 6552 | . 2 ⊢ (𝑋 RingHom 𝑌) = ( RingHom ‘〈𝑋, 𝑌〉) | |
10 | 4, 8, 9 | 3eqtr4g 2669 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 〈cop 4131 × cxp 5036 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 Ringcrg 18370 RingHom crh 18535 RngCatALTVcrngcALTV 41750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-res 5050 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: rhmsubcALTVlem3 41899 rhmsubcALTVlem4 41900 |
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