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Mirrors > Home > MPE Home > Th. List > drngid2 | Structured version Visualization version GIF version |
Description: Properties showing that an element 𝐼 is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.) |
Ref | Expression |
---|---|
drngid2.b | ⊢ 𝐵 = (Base‘𝑅) |
drngid2.t | ⊢ · = (.r‘𝑅) |
drngid2.o | ⊢ 0 = (0g‘𝑅) |
drngid2.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
drngid2 | ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1089 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ) ∧ (𝐼 · 𝐼) = 𝐼)) | |
2 | eldifsn 4811 | . . . . 5 ⊢ (𝐼 ∈ (𝐵 ∖ { 0 }) ↔ (𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 )) | |
3 | 2 | anbi1i 623 | . . . 4 ⊢ ((𝐼 ∈ (𝐵 ∖ { 0 }) ∧ (𝐼 · 𝐼) = 𝐼) ↔ ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ) ∧ (𝐼 · 𝐼) = 𝐼)) |
4 | 1, 3 | bitr4i 278 | . . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ (𝐼 ∈ (𝐵 ∖ { 0 }) ∧ (𝐼 · 𝐼) = 𝐼)) |
5 | drngid2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
6 | drngid2.o | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
7 | eqid 2740 | . . . . 5 ⊢ ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) | |
8 | 5, 6, 7 | drngmgp 20769 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp) |
9 | difss 4159 | . . . . . 6 ⊢ (𝐵 ∖ { 0 }) ⊆ 𝐵 | |
10 | eqid 2740 | . . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
11 | 10, 5 | mgpbas 20169 | . . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
12 | 7, 11 | ressbas2 17298 | . . . . . 6 ⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐵 → (𝐵 ∖ { 0 }) = (Base‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))) |
13 | 9, 12 | ax-mp 5 | . . . . 5 ⊢ (𝐵 ∖ { 0 }) = (Base‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) |
14 | 5 | fvexi 6936 | . . . . . 6 ⊢ 𝐵 ∈ V |
15 | difexg 5347 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 ∖ { 0 }) ∈ V) | |
16 | drngid2.t | . . . . . . . 8 ⊢ · = (.r‘𝑅) | |
17 | 10, 16 | mgpplusg 20167 | . . . . . . 7 ⊢ · = (+g‘(mulGrp‘𝑅)) |
18 | 7, 17 | ressplusg 17351 | . . . . . 6 ⊢ ((𝐵 ∖ { 0 }) ∈ V → · = (+g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))) |
19 | 14, 15, 18 | mp2b 10 | . . . . 5 ⊢ · = (+g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) |
20 | eqid 2740 | . . . . 5 ⊢ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) | |
21 | 13, 19, 20 | isgrpid2 19018 | . . . 4 ⊢ (((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp → ((𝐼 ∈ (𝐵 ∖ { 0 }) ∧ (𝐼 · 𝐼) = 𝐼) ↔ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = 𝐼)) |
22 | 8, 21 | syl 17 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ (𝐵 ∖ { 0 }) ∧ (𝐼 · 𝐼) = 𝐼) ↔ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = 𝐼)) |
23 | 4, 22 | bitrid 283 | . 2 ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = 𝐼)) |
24 | drngid2.u | . . . 4 ⊢ 1 = (1r‘𝑅) | |
25 | 5, 6, 24, 7 | drngid 20770 | . . 3 ⊢ (𝑅 ∈ DivRing → 1 = (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))) |
26 | 25 | eqeq1d 2742 | . 2 ⊢ (𝑅 ∈ DivRing → ( 1 = 𝐼 ↔ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = 𝐼)) |
27 | 23, 26 | bitr4d 282 | 1 ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ∖ cdif 3973 ⊆ wss 3976 {csn 4648 ‘cfv 6575 (class class class)co 7450 Basecbs 17260 ↾s cress 17289 +gcplusg 17313 .rcmulr 17314 0gc0g 17501 Grpcgrp 18975 mulGrpcmgp 20163 1rcur 20210 DivRingcdr 20753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-tpos 8269 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-0g 17503 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-grp 18978 df-minusg 18979 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20362 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-drng 20755 |
This theorem is referenced by: erng1r 40954 dvalveclem 40984 |
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