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Mirrors > Home > MPE Home > Th. List > srng1 | Structured version Visualization version GIF version |
Description: The conjugate of the ring identity is the identity. (This is sometimes taken as an axiom, and indeed the proof here follows because we defined *𝑟 to be a ring homomorphism, which preserves 1; nevertheless, it is redundant, as can be seen from the proof of issrngd 19253.) (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
srng1.i | ⊢ ∗ = (*𝑟‘𝑅) |
srng1.t | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
srng1 | ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 1 ) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srngring 19244 | . . 3 ⊢ (𝑅 ∈ *-Ring → 𝑅 ∈ Ring) | |
2 | eqid 2777 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | srng1.t | . . . 4 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | ringidcl 18955 | . . 3 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅)) |
5 | srng1.i | . . . 4 ⊢ ∗ = (*𝑟‘𝑅) | |
6 | eqid 2777 | . . . 4 ⊢ (*rf‘𝑅) = (*rf‘𝑅) | |
7 | 2, 5, 6 | stafval 19240 | . . 3 ⊢ ( 1 ∈ (Base‘𝑅) → ((*rf‘𝑅)‘ 1 ) = ( ∗ ‘ 1 )) |
8 | 1, 4, 7 | 3syl 18 | . 2 ⊢ (𝑅 ∈ *-Ring → ((*rf‘𝑅)‘ 1 ) = ( ∗ ‘ 1 )) |
9 | eqid 2777 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
10 | 9, 6 | srngrhm 19243 | . . 3 ⊢ (𝑅 ∈ *-Ring → (*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅))) |
11 | 9, 3 | oppr1 19021 | . . . 4 ⊢ 1 = (1r‘(oppr‘𝑅)) |
12 | 3, 11 | rhm1 19119 | . . 3 ⊢ ((*rf‘𝑅) ∈ (𝑅 RingHom (oppr‘𝑅)) → ((*rf‘𝑅)‘ 1 ) = 1 ) |
13 | 10, 12 | syl 17 | . 2 ⊢ (𝑅 ∈ *-Ring → ((*rf‘𝑅)‘ 1 ) = 1 ) |
14 | 8, 13 | eqtr3d 2815 | 1 ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 1 ) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 *𝑟cstv 16340 1rcur 18888 Ringcrg 18934 opprcoppr 19009 RingHom crh 19101 *rfcstf 19235 *-Ringcsr 19236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-mulr 16352 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-ghm 18042 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-rnghom 19104 df-staf 19237 df-srng 19238 |
This theorem is referenced by: (None) |
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