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Mirrors > Home > MPE Home > Th. List > drnginvrl | Structured version Visualization version GIF version |
Description: Property of the multiplicative inverse in a division ring. (recid2 11306 analog.) (Contributed by NM, 19-Apr-2014.) |
Ref | Expression |
---|---|
drnginvrl.b | ⊢ 𝐵 = (Base‘𝑅) |
drnginvrl.z | ⊢ 0 = (0g‘𝑅) |
drnginvrl.t | ⊢ · = (.r‘𝑅) |
drnginvrl.u | ⊢ 1 = (1r‘𝑅) |
drnginvrl.i | ⊢ 𝐼 = (invr‘𝑅) |
Ref | Expression |
---|---|
drnginvrl | ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drnginvrl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2820 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | drnginvrl.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
4 | 1, 2, 3 | drngunit 19502 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ (Unit‘𝑅) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
5 | drngring 19504 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
6 | drnginvrl.i | . . . . . 6 ⊢ 𝐼 = (invr‘𝑅) | |
7 | drnginvrl.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
8 | drnginvrl.u | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
9 | 2, 6, 7, 8 | unitlinv 19422 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → ((𝐼‘𝑋) · 𝑋) = 1 ) |
10 | 9 | ex 415 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑋 ∈ (Unit‘𝑅) → ((𝐼‘𝑋) · 𝑋) = 1 )) |
11 | 5, 10 | syl 17 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ (Unit‘𝑅) → ((𝐼‘𝑋) · 𝑋) = 1 )) |
12 | 4, 11 | sylbird 262 | . 2 ⊢ (𝑅 ∈ DivRing → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 )) |
13 | 12 | 3impib 1111 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 .rcmulr 16561 0gc0g 16708 1rcur 19246 Ringcrg 19292 Unitcui 19384 invrcinvr 19416 DivRingcdr 19497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-0g 16710 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-grp 18101 df-minusg 18102 df-mgp 19235 df-ur 19247 df-ring 19294 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-invr 19417 df-drng 19499 |
This theorem is referenced by: drngmul0or 19518 lvecvs0or 19875 lssvs0or 19877 lvecinv 19880 lspsnvs 19881 lspfixed 19895 lspsolv 19910 drngnidl 19997 matunitlindflem1 34923 lfl1 36239 eqlkr3 36270 lkrlsp 36271 tendolinv 38274 dochkr1 38647 dochkr1OLDN 38648 lclkrlem2m 38688 hdmapip1 39085 hgmapvvlem2 39093 |
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