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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ress1r | Structured version Visualization version GIF version |
Description: 1r is unaffected by restriction. This is a bit more generic than subrg1 18992. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
Ref | Expression |
---|---|
ress1r.s | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
ress1r.b | ⊢ 𝐵 = (Base‘𝑅) |
ress1r.1 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
ress1r | ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 1 = (1r‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ress1r.s | . . . 4 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
2 | ress1r.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 1, 2 | ressbas2 16133 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆)) |
4 | 3 | 3ad2ant3 1130 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘𝑆)) |
5 | simp3 1133 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
6 | fvex 6362 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
7 | 2, 6 | eqeltri 2835 | . . . 4 ⊢ 𝐵 ∈ V |
8 | ssexg 4956 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
9 | 5, 7, 8 | sylancl 697 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V) |
10 | eqid 2760 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 1, 10 | ressmulr 16208 | . . 3 ⊢ (𝐴 ∈ V → (.r‘𝑅) = (.r‘𝑆)) |
12 | 9, 11 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (.r‘𝑅) = (.r‘𝑆)) |
13 | simp2 1132 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 1 ∈ 𝐴) | |
14 | simpl1 1228 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ Ring) | |
15 | 5 | sselda 3744 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
16 | ress1r.1 | . . . 4 ⊢ 1 = (1r‘𝑅) | |
17 | 2, 10, 16 | ringlidm 18771 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑥) = 𝑥) |
18 | 14, 15, 17 | syl2anc 696 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → ( 1 (.r‘𝑅)𝑥) = 𝑥) |
19 | 2, 10, 16 | ringridm 18772 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝑅) 1 ) = 𝑥) |
20 | 14, 15, 19 | syl2anc 696 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(.r‘𝑅) 1 ) = 𝑥) |
21 | 4, 12, 13, 18, 20 | rngurd 30097 | 1 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 1 = (1r‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ⊆ wss 3715 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 ↾s cress 16060 .rcmulr 16144 1rcur 18701 Ringcrg 18747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-0g 16304 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mgp 18690 df-ur 18702 df-ring 18749 |
This theorem is referenced by: xrge0slmod 30153 |
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