Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19447 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿)) |
5 | 4 | breq2d 5080 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐾) ↔ 𝑧(∥r‘𝐾)(1r‘𝐿))) |
6 | 4 | breq2d 5080 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾) ↔ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿))) |
7 | 5, 6 | anbi12d 632 | . . . 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 19448 | . . . . . 6 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿)) |
9 | 8 | breqd 5079 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐿) ↔ 𝑧(∥r‘𝐿)(1r‘𝐿))) |
10 | eqid 2823 | . . . . . . . . 9 ⊢ (oppr‘𝐾) = (oppr‘𝐾) | |
11 | eqid 2823 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | 10, 11 | opprbas 19381 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘(oppr‘𝐾)) |
13 | 1, 12 | syl6eq 2874 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐾))) |
14 | eqid 2823 | . . . . . . . . 9 ⊢ (oppr‘𝐿) = (oppr‘𝐿) | |
15 | eqid 2823 | . . . . . . . . 9 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
16 | 14, 15 | opprbas 19381 | . . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘(oppr‘𝐿)) |
17 | 2, 16 | syl6eq 2874 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐿))) |
18 | 3 | ancom2s 648 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
19 | eqid 2823 | . . . . . . . . 9 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
20 | eqid 2823 | . . . . . . . . 9 ⊢ (.r‘(oppr‘𝐾)) = (.r‘(oppr‘𝐾)) | |
21 | 11, 19, 10, 20 | opprmul 19378 | . . . . . . . 8 ⊢ (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦) |
22 | eqid 2823 | . . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
23 | eqid 2823 | . . . . . . . . 9 ⊢ (.r‘(oppr‘𝐿)) = (.r‘(oppr‘𝐿)) | |
24 | 15, 22, 14, 23 | opprmul 19378 | . . . . . . . 8 ⊢ (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦) |
25 | 18, 21, 24 | 3eqtr4g 2883 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑦(.r‘(oppr‘𝐿))𝑥)) |
26 | 13, 17, 25 | dvdsrpropd 19448 | . . . . . 6 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐿))) |
27 | 26 | breqd 5079 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿) ↔ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))) |
28 | 9, 27 | anbi12d 632 | . . . 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 281 | . . 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 2823 | . . . 4 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
31 | eqid 2823 | . . . 4 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
32 | eqid 2823 | . . . 4 ⊢ (∥r‘𝐾) = (∥r‘𝐾) | |
33 | eqid 2823 | . . . 4 ⊢ (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐾)) | |
34 | 30, 31, 32, 10, 33 | isunit 19409 | . . 3 ⊢ (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾))) |
35 | eqid 2823 | . . . 4 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
36 | eqid 2823 | . . . 4 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
37 | eqid 2823 | . . . 4 ⊢ (∥r‘𝐿) = (∥r‘𝐿) | |
38 | eqid 2823 | . . . 4 ⊢ (∥r‘(oppr‘𝐿)) = (∥r‘(oppr‘𝐿)) | |
39 | 35, 36, 37, 14, 38 | isunit 19409 | . . 3 ⊢ (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r‘𝐿)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))) |
40 | 29, 34, 39 | 3bitr4g 316 | . 2 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿))) |
41 | 40 | eqrdv 2821 | 1 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 1rcur 19253 opprcoppr 19374 ∥rcdsr 19390 Unitcui 19391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mgp 19242 df-ur 19254 df-oppr 19375 df-dvdsr 19393 df-unit 19394 |
This theorem is referenced by: invrpropd 19450 drngprop 19515 drngpropd 19531 sradrng 30990 |
Copyright terms: Public domain | W3C validator |