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 19685 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿)) |
5 | 4 | breq2d 5055 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐾) ↔ 𝑧(∥r‘𝐾)(1r‘𝐿))) |
6 | 4 | breq2d 5055 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾) ↔ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿))) |
7 | 5, 6 | anbi12d 634 | . . . 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 19686 | . . . . . 6 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿)) |
9 | 8 | breqd 5054 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐿) ↔ 𝑧(∥r‘𝐿)(1r‘𝐿))) |
10 | eqid 2734 | . . . . . . . . 9 ⊢ (oppr‘𝐾) = (oppr‘𝐾) | |
11 | eqid 2734 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | 10, 11 | opprbas 19619 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘(oppr‘𝐾)) |
13 | 1, 12 | eqtrdi 2790 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐾))) |
14 | eqid 2734 | . . . . . . . . 9 ⊢ (oppr‘𝐿) = (oppr‘𝐿) | |
15 | eqid 2734 | . . . . . . . . 9 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
16 | 14, 15 | opprbas 19619 | . . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘(oppr‘𝐿)) |
17 | 2, 16 | eqtrdi 2790 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐿))) |
18 | 3 | ancom2s 650 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
19 | eqid 2734 | . . . . . . . . 9 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
20 | eqid 2734 | . . . . . . . . 9 ⊢ (.r‘(oppr‘𝐾)) = (.r‘(oppr‘𝐾)) | |
21 | 11, 19, 10, 20 | opprmul 19616 | . . . . . . . 8 ⊢ (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦) |
22 | eqid 2734 | . . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
23 | eqid 2734 | . . . . . . . . 9 ⊢ (.r‘(oppr‘𝐿)) = (.r‘(oppr‘𝐿)) | |
24 | 15, 22, 14, 23 | opprmul 19616 | . . . . . . . 8 ⊢ (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦) |
25 | 18, 21, 24 | 3eqtr4g 2799 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑦(.r‘(oppr‘𝐿))𝑥)) |
26 | 13, 17, 25 | dvdsrpropd 19686 | . . . . . 6 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐿))) |
27 | 26 | breqd 5054 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿) ↔ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))) |
28 | 9, 27 | anbi12d 634 | . . . 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 282 | . . 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 2734 | . . . 4 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
31 | eqid 2734 | . . . 4 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
32 | eqid 2734 | . . . 4 ⊢ (∥r‘𝐾) = (∥r‘𝐾) | |
33 | eqid 2734 | . . . 4 ⊢ (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐾)) | |
34 | 30, 31, 32, 10, 33 | isunit 19647 | . . 3 ⊢ (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾))) |
35 | eqid 2734 | . . . 4 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
36 | eqid 2734 | . . . 4 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
37 | eqid 2734 | . . . 4 ⊢ (∥r‘𝐿) = (∥r‘𝐿) | |
38 | eqid 2734 | . . . 4 ⊢ (∥r‘(oppr‘𝐿)) = (∥r‘(oppr‘𝐿)) | |
39 | 35, 36, 37, 14, 38 | isunit 19647 | . . 3 ⊢ (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r‘𝐿)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))) |
40 | 29, 34, 39 | 3bitr4g 317 | . 2 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿))) |
41 | 40 | eqrdv 2732 | 1 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 .rcmulr 16768 1rcur 19488 opprcoppr 19612 ∥rcdsr 19628 Unitcui 19629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-plusg 16780 df-mulr 16781 df-0g 16918 df-mgp 19477 df-ur 19489 df-oppr 19613 df-dvdsr 19631 df-unit 19632 |
This theorem is referenced by: invrpropd 19688 drngprop 19750 drngpropd 19766 sradrng 31359 |
Copyright terms: Public domain | W3C validator |