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Mirrors > Home > MPE Home > Th. List > unitpropd | Structured version Visualization version GIF version |
Description: The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) |
Ref | Expression |
---|---|
rngidpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
rngidpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
rngidpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
unitpropd | ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngidpropd.1 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | rngidpropd.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | rngidpropd.3 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
4 | 1, 2, 3 | rngidpropd 19135 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿)) |
5 | 4 | breq2d 4974 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐾) ↔ 𝑧(∥r‘𝐾)(1r‘𝐿))) |
6 | 4 | breq2d 4974 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾) ↔ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿))) |
7 | 5, 6 | anbi12d 630 | . . . 4 ⊢ (𝜑 → ((𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾)) ↔ (𝑧(∥r‘𝐾)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿)))) |
8 | 1, 2, 3 | dvdsrpropd 19136 | . . . . . 6 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿)) |
9 | 8 | breqd 4973 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐿) ↔ 𝑧(∥r‘𝐿)(1r‘𝐿))) |
10 | eqid 2795 | . . . . . . . . 9 ⊢ (oppr‘𝐾) = (oppr‘𝐾) | |
11 | eqid 2795 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | 10, 11 | opprbas 19069 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘(oppr‘𝐾)) |
13 | 1, 12 | syl6eq 2847 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐾))) |
14 | eqid 2795 | . . . . . . . . 9 ⊢ (oppr‘𝐿) = (oppr‘𝐿) | |
15 | eqid 2795 | . . . . . . . . 9 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
16 | 14, 15 | opprbas 19069 | . . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘(oppr‘𝐿)) |
17 | 2, 16 | syl6eq 2847 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐿))) |
18 | 3 | ancom2s 646 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
19 | eqid 2795 | . . . . . . . . 9 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
20 | eqid 2795 | . . . . . . . . 9 ⊢ (.r‘(oppr‘𝐾)) = (.r‘(oppr‘𝐾)) | |
21 | 11, 19, 10, 20 | opprmul 19066 | . . . . . . . 8 ⊢ (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦) |
22 | eqid 2795 | . . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
23 | eqid 2795 | . . . . . . . . 9 ⊢ (.r‘(oppr‘𝐿)) = (.r‘(oppr‘𝐿)) | |
24 | 15, 22, 14, 23 | opprmul 19066 | . . . . . . . 8 ⊢ (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦) |
25 | 18, 21, 24 | 3eqtr4g 2856 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑦(.r‘(oppr‘𝐿))𝑥)) |
26 | 13, 17, 25 | dvdsrpropd 19136 | . . . . . 6 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐿))) |
27 | 26 | breqd 4973 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿) ↔ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))) |
28 | 9, 27 | anbi12d 630 | . . . 4 ⊢ (𝜑 → ((𝑧(∥r‘𝐾)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿)) ↔ (𝑧(∥r‘𝐿)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿)))) |
29 | 7, 28 | bitrd 280 | . . 3 ⊢ (𝜑 → ((𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾)) ↔ (𝑧(∥r‘𝐿)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿)))) |
30 | eqid 2795 | . . . 4 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
31 | eqid 2795 | . . . 4 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
32 | eqid 2795 | . . . 4 ⊢ (∥r‘𝐾) = (∥r‘𝐾) | |
33 | eqid 2795 | . . . 4 ⊢ (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐾)) | |
34 | 30, 31, 32, 10, 33 | isunit 19097 | . . 3 ⊢ (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾))) |
35 | eqid 2795 | . . . 4 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
36 | eqid 2795 | . . . 4 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
37 | eqid 2795 | . . . 4 ⊢ (∥r‘𝐿) = (∥r‘𝐿) | |
38 | eqid 2795 | . . . 4 ⊢ (∥r‘(oppr‘𝐿)) = (∥r‘(oppr‘𝐿)) | |
39 | 35, 36, 37, 14, 38 | isunit 19097 | . . 3 ⊢ (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r‘𝐿)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))) |
40 | 29, 34, 39 | 3bitr4g 315 | . 2 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿))) |
41 | 40 | eqrdv 2793 | 1 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 class class class wbr 4962 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 .rcmulr 16395 1rcur 18941 opprcoppr 19062 ∥rcdsr 19078 Unitcui 19079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-plusg 16407 df-mulr 16408 df-0g 16544 df-mgp 18930 df-ur 18942 df-oppr 19063 df-dvdsr 19081 df-unit 19082 |
This theorem is referenced by: invrpropd 19138 drngprop 19203 drngpropd 19219 sradrng 30592 |
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