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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for rhmsubc 45648. (Contributed by AV, 2-Mar-2020.) |
Ref | Expression |
---|---|
rngcrescrhm.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcrescrhm.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rngcrescrhm.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) |
rngcrescrhm.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) |
Ref | Expression |
---|---|
rhmsubclem2 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5626 | . . . 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 6794 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) = ( RingHom ‘〈𝑋, 𝑌〉)) |
4 | df-ov 7278 | . . 3 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
5 | rngcrescrhm.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
6 | 5 | fveq1i 6775 | . . 3 ⊢ (𝐻‘〈𝑋, 𝑌〉) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) |
7 | 4, 6 | eqtri 2766 | . 2 ⊢ (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘〈𝑋, 𝑌〉) |
8 | df-ov 7278 | . 2 ⊢ (𝑋 RingHom 𝑌) = ( RingHom ‘〈𝑋, 𝑌〉) | |
9 | 3, 7, 8 | 3eqtr4g 2803 | 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 2106 ∩ cin 3886 〈cop 4567 × cxp 5587 ↾ cres 5591 ‘cfv 6433 (class class class)co 7275 Ringcrg 19783 RingHom crh 19956 RngCatcrngc 45515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-res 5601 df-iota 6391 df-fv 6441 df-ov 7278 |
This theorem is referenced by: rhmsubclem3 45646 rhmsubclem4 45647 |
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