![]() |
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 20313 | . . . . . 6 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿)) |
5 | 4 | breq2d 5160 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐾) ↔ 𝑧(∥r‘𝐾)(1r‘𝐿))) |
6 | 4 | breq2d 5160 | . . . . 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 20314 | . . . . . 6 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿)) |
9 | 8 | breqd 5159 | . . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐿) ↔ 𝑧(∥r‘𝐿)(1r‘𝐿))) |
10 | eqid 2731 | . . . . . . . . 9 ⊢ (oppr‘𝐾) = (oppr‘𝐾) | |
11 | eqid 2731 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | 10, 11 | opprbas 20239 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘(oppr‘𝐾)) |
13 | 1, 12 | eqtrdi 2787 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐾))) |
14 | eqid 2731 | . . . . . . . . 9 ⊢ (oppr‘𝐿) = (oppr‘𝐿) | |
15 | eqid 2731 | . . . . . . . . 9 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
16 | 14, 15 | opprbas 20239 | . . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘(oppr‘𝐿)) |
17 | 2, 16 | eqtrdi 2787 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐿))) |
18 | 3 | ancom2s 647 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
19 | eqid 2731 | . . . . . . . . 9 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
20 | eqid 2731 | . . . . . . . . 9 ⊢ (.r‘(oppr‘𝐾)) = (.r‘(oppr‘𝐾)) | |
21 | 11, 19, 10, 20 | opprmul 20235 | . . . . . . . 8 ⊢ (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦) |
22 | eqid 2731 | . . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
23 | eqid 2731 | . . . . . . . . 9 ⊢ (.r‘(oppr‘𝐿)) = (.r‘(oppr‘𝐿)) | |
24 | 15, 22, 14, 23 | opprmul 20235 | . . . . . . . 8 ⊢ (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦) |
25 | 18, 21, 24 | 3eqtr4g 2796 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑦(.r‘(oppr‘𝐿))𝑥)) |
26 | 13, 17, 25 | dvdsrpropd 20314 | . . . . . 6 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐿))) |
27 | 26 | breqd 5159 | . . . . 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 279 | . . 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 2731 | . . . 4 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
31 | eqid 2731 | . . . 4 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
32 | eqid 2731 | . . . 4 ⊢ (∥r‘𝐾) = (∥r‘𝐾) | |
33 | eqid 2731 | . . . 4 ⊢ (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐾)) | |
34 | 30, 31, 32, 10, 33 | isunit 20271 | . . 3 ⊢ (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾))) |
35 | eqid 2731 | . . . 4 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
36 | eqid 2731 | . . . 4 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
37 | eqid 2731 | . . . 4 ⊢ (∥r‘𝐿) = (∥r‘𝐿) | |
38 | eqid 2731 | . . . 4 ⊢ (∥r‘(oppr‘𝐿)) = (∥r‘(oppr‘𝐿)) | |
39 | 35, 36, 37, 14, 38 | isunit 20271 | . . 3 ⊢ (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r‘𝐿)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))) |
40 | 29, 34, 39 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿))) |
41 | 40 | eqrdv 2729 | 1 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 .rcmulr 17205 1rcur 20082 opprcoppr 20231 ∥rcdsr 20252 Unitcui 20253 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-0g 17394 df-mgp 20036 df-ur 20083 df-oppr 20232 df-dvdsr 20255 df-unit 20256 |
This theorem is referenced by: invrpropd 20316 drngprop 20598 drngpropd 20620 sradrng 33124 |
Copyright terms: Public domain | W3C validator |