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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 20366. (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 17179 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑆)) |
4 | 3 | 3ad2ant3 1136 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘𝑆)) |
5 | simp3 1139 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
6 | 2 | fvexi 6903 | . . . 4 ⊢ 𝐵 ∈ V |
7 | ssexg 5323 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
8 | 5, 6, 7 | sylancl 587 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V) |
9 | eqid 2733 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | 1, 9 | ressmulr 17249 | . . 3 ⊢ (𝐴 ∈ V → (.r‘𝑅) = (.r‘𝑆)) |
11 | 8, 10 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (.r‘𝑅) = (.r‘𝑆)) |
12 | simp2 1138 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 1 ∈ 𝐴) | |
13 | simpl1 1192 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ Ring) | |
14 | 5 | sselda 3982 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
15 | ress1r.1 | . . . 4 ⊢ 1 = (1r‘𝑅) | |
16 | 2, 9, 15 | ringlidm 20080 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ( 1 (.r‘𝑅)𝑥) = 𝑥) |
17 | 13, 14, 16 | syl2anc 585 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → ( 1 (.r‘𝑅)𝑥) = 𝑥) |
18 | 2, 9, 15 | ringridm 20081 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝑅) 1 ) = 𝑥) |
19 | 13, 14, 18 | syl2anc 585 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥(.r‘𝑅) 1 ) = 𝑥) |
20 | 4, 11, 12, 17, 19 | rngurd 32368 | 1 ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 1 = (1r‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3948 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 ↾s cress 17170 .rcmulr 17195 1rcur 19999 Ringcrg 20050 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-0g 17384 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mgp 19983 df-ur 20000 df-ring 20052 |
This theorem is referenced by: xrge0slmod 32452 ply1annnr 32753 |
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