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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 1039 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ) ∧ (𝐼 · 𝐼) = 𝐼)) | |
2 | eldifsn 4317 | . . . . 5 ⊢ (𝐼 ∈ (𝐵 ∖ { 0 }) ↔ (𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 )) | |
3 | 2 | anbi1i 731 | . . . 4 ⊢ ((𝐼 ∈ (𝐵 ∖ { 0 }) ∧ (𝐼 · 𝐼) = 𝐼) ↔ ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ) ∧ (𝐼 · 𝐼) = 𝐼)) |
4 | 1, 3 | bitr4i 267 | . . 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 2622 | . . . . 5 ⊢ ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) | |
8 | 5, 6, 7 | drngmgp 18759 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp) |
9 | difss 3737 | . . . . . 6 ⊢ (𝐵 ∖ { 0 }) ⊆ 𝐵 | |
10 | eqid 2622 | . . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
11 | 10, 5 | mgpbas 18495 | . . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
12 | 7, 11 | ressbas2 15931 | . . . . . 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 | fvex 6201 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
15 | 5, 14 | eqeltri 2697 | . . . . . 6 ⊢ 𝐵 ∈ V |
16 | difexg 4808 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 ∖ { 0 }) ∈ V) | |
17 | drngid2.t | . . . . . . . 8 ⊢ · = (.r‘𝑅) | |
18 | 10, 17 | mgpplusg 18493 | . . . . . . 7 ⊢ · = (+g‘(mulGrp‘𝑅)) |
19 | 7, 18 | ressplusg 15993 | . . . . . 6 ⊢ ((𝐵 ∖ { 0 }) ∈ V → · = (+g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))) |
20 | 15, 16, 19 | mp2b 10 | . . . . 5 ⊢ · = (+g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) |
21 | eqid 2622 | . . . . 5 ⊢ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) | |
22 | 13, 20, 21 | isgrpid2 17458 | . . . 4 ⊢ (((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp → ((𝐼 ∈ (𝐵 ∖ { 0 }) ∧ (𝐼 · 𝐼) = 𝐼) ↔ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = 𝐼)) |
23 | 8, 22 | syl 17 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ (𝐵 ∖ { 0 }) ∧ (𝐼 · 𝐼) = 𝐼) ↔ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = 𝐼)) |
24 | 4, 23 | syl5bb 272 | . 2 ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = 𝐼)) |
25 | drngid2.u | . . . 4 ⊢ 1 = (1r‘𝑅) | |
26 | 5, 6, 25, 7 | drngid 18761 | . . 3 ⊢ (𝑅 ∈ DivRing → 1 = (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))) |
27 | 26 | eqeq1d 2624 | . 2 ⊢ (𝑅 ∈ DivRing → ( 1 = 𝐼 ↔ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = 𝐼)) |
28 | 24, 27 | bitr4d 271 | 1 ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 +gcplusg 15941 .rcmulr 15942 0gc0g 16100 Grpcgrp 17422 mulGrpcmgp 18489 1rcur 18501 DivRingcdr 18747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 |
This theorem is referenced by: erng1r 36283 dvalveclem 36314 |
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