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Mirrors > Home > MPE Home > Th. List > drngpropd | Structured version Visualization version GIF version |
Description: If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.) |
Ref | Expression |
---|---|
drngpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
drngpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
drngpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
drngpropd.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
drngpropd | ⊢ (𝜑 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngpropd.1 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | drngpropd.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | drngpropd.4 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
4 | 1, 2, 3 | unitpropd 18697 | . . . . . 6 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
5 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Unit‘𝐾) = (Unit‘𝐿)) |
6 | 1, 2 | eqtr3d 2658 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
7 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Base‘𝐾) = (Base‘𝐿)) |
8 | 1 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘𝐾)) |
9 | 2 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘𝐿)) |
10 | drngpropd.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
11 | 10 | adantlr 751 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
12 | 8, 9, 11 | grpidpropd 17261 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (0g‘𝐾) = (0g‘𝐿)) |
13 | 12 | sneqd 4189 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → {(0g‘𝐾)} = {(0g‘𝐿)}) |
14 | 7, 13 | difeq12d 3729 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((Base‘𝐾) ∖ {(0g‘𝐾)}) = ((Base‘𝐿) ∖ {(0g‘𝐿)})) |
15 | 5, 14 | eqeq12d 2637 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}) ↔ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)}))) |
16 | 15 | pm5.32da 673 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
17 | 1, 2, 10, 3 | ringpropd 18582 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
18 | 17 | anbi1d 741 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
19 | 16, 18 | bitrd 268 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
20 | eqid 2622 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2622 | . . 3 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
22 | eqid 2622 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
23 | 20, 21, 22 | isdrng 18751 | . 2 ⊢ (𝐾 ∈ DivRing ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}))) |
24 | eqid 2622 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
25 | eqid 2622 | . . 3 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
26 | eqid 2622 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
27 | 24, 25, 26 | isdrng 18751 | . 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)}))) |
28 | 19, 23, 27 | 3bitr4g 303 | 1 ⊢ (𝜑 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 {csn 4177 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 0gc0g 16100 Ringcrg 18547 Unitcui 18639 DivRingcdr 18747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-drng 18749 |
This theorem is referenced by: fldpropd 18775 lvecprop2d 19166 hlhildrng 37244 |
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