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Mirrors > Home > MPE Home > Th. List > opprunit | Structured version Visualization version GIF version |
Description: Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
opprunit.1 | ⊢ 𝑈 = (Unit‘𝑅) |
opprunit.2 | ⊢ 𝑆 = (oppr‘𝑅) |
Ref | Expression |
---|---|
opprunit | ⊢ 𝑈 = (Unit‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprunit.2 | . . . . . . . . . . 11 ⊢ 𝑆 = (oppr‘𝑅) | |
2 | eqid 2824 | . . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | opprbas 19382 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑆) |
4 | eqid 2824 | . . . . . . . . . 10 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
5 | eqid 2824 | . . . . . . . . . 10 ⊢ (oppr‘𝑆) = (oppr‘𝑆) | |
6 | eqid 2824 | . . . . . . . . . 10 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆)) | |
7 | 3, 4, 5, 6 | opprmul 19379 | . . . . . . . . 9 ⊢ (𝑦(.r‘(oppr‘𝑆))𝑥) = (𝑥(.r‘𝑆)𝑦) |
8 | eqid 2824 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
9 | 2, 8, 1, 4 | opprmul 19379 | . . . . . . . . 9 ⊢ (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑅)𝑥) |
10 | 7, 9 | eqtr2i 2848 | . . . . . . . 8 ⊢ (𝑦(.r‘𝑅)𝑥) = (𝑦(.r‘(oppr‘𝑆))𝑥) |
11 | 10 | eqeq1i 2829 | . . . . . . 7 ⊢ ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ↔ (𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅)) |
12 | 11 | rexbii 3250 | . . . . . 6 ⊢ (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅)) |
13 | 12 | anbi2i 624 | . . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅))) |
14 | eqid 2824 | . . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
15 | 2, 14, 8 | dvdsr 19399 | . . . . 5 ⊢ (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))) |
16 | 5, 3 | opprbas 19382 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑆)) |
17 | eqid 2824 | . . . . . 6 ⊢ (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)) | |
18 | 16, 17, 6 | dvdsr 19399 | . . . . 5 ⊢ (𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅))) |
19 | 13, 15, 18 | 3bitr4i 305 | . . . 4 ⊢ (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅)) |
20 | 19 | anbi2ci 626 | . . 3 ⊢ ((𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘𝑆)(1r‘𝑅)) ↔ (𝑥(∥r‘𝑆)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅))) |
21 | opprunit.1 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
22 | eqid 2824 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
23 | eqid 2824 | . . . 4 ⊢ (∥r‘𝑆) = (∥r‘𝑆) | |
24 | 21, 22, 14, 1, 23 | isunit 19410 | . . 3 ⊢ (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘𝑆)(1r‘𝑅))) |
25 | eqid 2824 | . . . 4 ⊢ (Unit‘𝑆) = (Unit‘𝑆) | |
26 | 1, 22 | oppr1 19387 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑆) |
27 | 25, 26, 23, 5, 17 | isunit 19410 | . . 3 ⊢ (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥(∥r‘𝑆)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅))) |
28 | 20, 24, 27 | 3bitr4i 305 | . 2 ⊢ (𝑥 ∈ 𝑈 ↔ 𝑥 ∈ (Unit‘𝑆)) |
29 | 28 | eqriv 2821 | 1 ⊢ 𝑈 = (Unit‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 .rcmulr 16569 1rcur 19254 opprcoppr 19375 ∥rcdsr 19391 Unitcui 19392 |
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 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mgp 19243 df-ur 19255 df-oppr 19376 df-dvdsr 19394 df-unit 19395 |
This theorem is referenced by: opprirred 19455 irredlmul 19461 opprdrng 19529 ply1divalg2 24735 |
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