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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 | β’ (π β π΅ = (BaseβπΎ)) |
drngpropd.2 | β’ (π β π΅ = (BaseβπΏ)) |
drngpropd.3 | β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) |
drngpropd.4 | β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(.rβπΎ)π¦) = (π₯(.rβπΏ)π¦)) |
Ref | Expression |
---|---|
drngpropd | β’ (π β (πΎ β DivRing β πΏ β DivRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngpropd.1 | . . . . . . 7 β’ (π β π΅ = (BaseβπΎ)) | |
2 | drngpropd.2 | . . . . . . 7 β’ (π β π΅ = (BaseβπΏ)) | |
3 | drngpropd.4 | . . . . . . 7 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(.rβπΎ)π¦) = (π₯(.rβπΏ)π¦)) | |
4 | 1, 2, 3 | unitpropd 20309 | . . . . . 6 β’ (π β (UnitβπΎ) = (UnitβπΏ)) |
5 | 4 | adantr 480 | . . . . 5 β’ ((π β§ πΎ β Ring) β (UnitβπΎ) = (UnitβπΏ)) |
6 | 1, 2 | eqtr3d 2773 | . . . . . . 7 β’ (π β (BaseβπΎ) = (BaseβπΏ)) |
7 | 6 | adantr 480 | . . . . . 6 β’ ((π β§ πΎ β Ring) β (BaseβπΎ) = (BaseβπΏ)) |
8 | 1 | adantr 480 | . . . . . . . 8 β’ ((π β§ πΎ β Ring) β π΅ = (BaseβπΎ)) |
9 | 2 | adantr 480 | . . . . . . . 8 β’ ((π β§ πΎ β Ring) β π΅ = (BaseβπΏ)) |
10 | drngpropd.3 | . . . . . . . . 9 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) | |
11 | 10 | adantlr 712 | . . . . . . . 8 β’ (((π β§ πΎ β Ring) β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) |
12 | 8, 9, 11 | grpidpropd 18588 | . . . . . . 7 β’ ((π β§ πΎ β Ring) β (0gβπΎ) = (0gβπΏ)) |
13 | 12 | sneqd 4641 | . . . . . 6 β’ ((π β§ πΎ β Ring) β {(0gβπΎ)} = {(0gβπΏ)}) |
14 | 7, 13 | difeq12d 4124 | . . . . 5 β’ ((π β§ πΎ β Ring) β ((BaseβπΎ) β {(0gβπΎ)}) = ((BaseβπΏ) β {(0gβπΏ)})) |
15 | 5, 14 | eqeq12d 2747 | . . . 4 β’ ((π β§ πΎ β Ring) β ((UnitβπΎ) = ((BaseβπΎ) β {(0gβπΎ)}) β (UnitβπΏ) = ((BaseβπΏ) β {(0gβπΏ)}))) |
16 | 15 | pm5.32da 578 | . . 3 β’ (π β ((πΎ β Ring β§ (UnitβπΎ) = ((BaseβπΎ) β {(0gβπΎ)})) β (πΎ β Ring β§ (UnitβπΏ) = ((BaseβπΏ) β {(0gβπΏ)})))) |
17 | 1, 2, 10, 3 | ringpropd 20177 | . . . 4 β’ (π β (πΎ β Ring β πΏ β Ring)) |
18 | 17 | anbi1d 629 | . . 3 β’ (π β ((πΎ β Ring β§ (UnitβπΏ) = ((BaseβπΏ) β {(0gβπΏ)})) β (πΏ β Ring β§ (UnitβπΏ) = ((BaseβπΏ) β {(0gβπΏ)})))) |
19 | 16, 18 | bitrd 278 | . 2 β’ (π β ((πΎ β Ring β§ (UnitβπΎ) = ((BaseβπΎ) β {(0gβπΎ)})) β (πΏ β Ring β§ (UnitβπΏ) = ((BaseβπΏ) β {(0gβπΏ)})))) |
20 | eqid 2731 | . . 3 β’ (BaseβπΎ) = (BaseβπΎ) | |
21 | eqid 2731 | . . 3 β’ (UnitβπΎ) = (UnitβπΎ) | |
22 | eqid 2731 | . . 3 β’ (0gβπΎ) = (0gβπΎ) | |
23 | 20, 21, 22 | isdrng 20505 | . 2 β’ (πΎ β DivRing β (πΎ β Ring β§ (UnitβπΎ) = ((BaseβπΎ) β {(0gβπΎ)}))) |
24 | eqid 2731 | . . 3 β’ (BaseβπΏ) = (BaseβπΏ) | |
25 | eqid 2731 | . . 3 β’ (UnitβπΏ) = (UnitβπΏ) | |
26 | eqid 2731 | . . 3 β’ (0gβπΏ) = (0gβπΏ) | |
27 | 24, 25, 26 | isdrng 20505 | . 2 β’ (πΏ β DivRing β (πΏ β Ring β§ (UnitβπΏ) = ((BaseβπΏ) β {(0gβπΏ)}))) |
28 | 19, 23, 27 | 3bitr4g 313 | 1 β’ (π β (πΎ β DivRing β πΏ β DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 β cdif 3946 {csn 4629 βcfv 6544 (class class class)co 7412 Basecbs 17149 +gcplusg 17202 .rcmulr 17203 0gc0g 17390 Ringcrg 20128 Unitcui 20247 DivRingcdr 20501 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-2nd 7979 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-mgp 20030 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-drng 20503 |
This theorem is referenced by: fldpropd 20539 lvecprop2d 20925 hlhildrng 41131 |
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