![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2734 | . . . . 5 ⊢ ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) | |
8 | 5, 6, 7 | drngmgp 20762 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp) |
9 | difss 4153 | . . . . . 6 ⊢ (𝐵 ∖ { 0 }) ⊆ 𝐵 | |
10 | eqid 2734 | . . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
11 | 10, 5 | mgpbas 20162 | . . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
12 | 7, 11 | ressbas2 17291 | . . . . . 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 6933 | . . . . . 6 ⊢ 𝐵 ∈ V |
15 | difexg 5350 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 ∖ { 0 }) ∈ V) | |
16 | drngid2.t | . . . . . . . 8 ⊢ · = (.r‘𝑅) | |
17 | 10, 16 | mgpplusg 20160 | . . . . . . 7 ⊢ · = (+g‘(mulGrp‘𝑅)) |
18 | 7, 17 | ressplusg 17344 | . . . . . 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 2734 | . . . . 5 ⊢ (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) = (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))) | |
21 | 13, 19, 20 | isgrpid2 19011 | . . . 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 20763 | . . 3 ⊢ (𝑅 ∈ DivRing → 1 = (0g‘((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })))) |
26 | 25 | eqeq1d 2736 | . 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 2103 ≠ wne 2942 Vcvv 3482 ∖ cdif 3967 ⊆ wss 3970 {csn 4648 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 ↾s cress 17282 +gcplusg 17306 .rcmulr 17307 0gc0g 17494 Grpcgrp 18968 mulGrpcmgp 20156 1rcur 20203 DivRingcdr 20746 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-0g 17496 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-grp 18971 df-minusg 18972 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-drng 20748 |
This theorem is referenced by: erng1r 40900 dvalveclem 40930 |
Copyright terms: Public domain | W3C validator |