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Mirrors > Home > MPE Home > Th. List > Mathboxes > riccrng1 | Structured version Visualization version GIF version |
Description: Ring isomorphism preserves (multiplicative) commutativity. (Contributed by SN, 10-Jan-2025.) |
Ref | Expression |
---|---|
riccrng1 | β’ ((π βπ π β§ π β CRing) β π β CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brric 20396 | . . 3 β’ (π βπ π β (π RingIso π) β β ) | |
2 | n0 4346 | . . 3 β’ ((π RingIso π) β β β βπ π β (π RingIso π)) | |
3 | 1, 2 | bitri 275 | . 2 β’ (π βπ π β βπ π β (π RingIso π)) |
4 | eqid 2731 | . . . . . . . . . . 11 β’ (Baseβπ ) = (Baseβπ ) | |
5 | eqid 2731 | . . . . . . . . . . 11 β’ (Baseβπ) = (Baseβπ) | |
6 | 4, 5 | rimf1o 20386 | . . . . . . . . . 10 β’ (π β (π RingIso π) β π:(Baseβπ )β1-1-ontoβ(Baseβπ)) |
7 | f1ofo 6840 | . . . . . . . . . 10 β’ (π:(Baseβπ )β1-1-ontoβ(Baseβπ) β π:(Baseβπ )βontoβ(Baseβπ)) | |
8 | foima 6810 | . . . . . . . . . 10 β’ (π:(Baseβπ )βontoβ(Baseβπ) β (π β (Baseβπ )) = (Baseβπ)) | |
9 | 6, 7, 8 | 3syl 18 | . . . . . . . . 9 β’ (π β (π RingIso π) β (π β (Baseβπ )) = (Baseβπ)) |
10 | 9 | oveq2d 7428 | . . . . . . . 8 β’ (π β (π RingIso π) β (π βΎs (π β (Baseβπ ))) = (π βΎs (Baseβπ))) |
11 | rimrcl2 41396 | . . . . . . . . 9 β’ (π β (π RingIso π) β π β Ring) | |
12 | 5 | ressid 17194 | . . . . . . . . 9 β’ (π β Ring β (π βΎs (Baseβπ)) = π) |
13 | 11, 12 | syl 17 | . . . . . . . 8 β’ (π β (π RingIso π) β (π βΎs (Baseβπ)) = π) |
14 | 10, 13 | eqtr2d 2772 | . . . . . . 7 β’ (π β (π RingIso π) β π = (π βΎs (π β (Baseβπ )))) |
15 | 14 | adantr 480 | . . . . . 6 β’ ((π β (π RingIso π) β§ π β CRing) β π = (π βΎs (π β (Baseβπ )))) |
16 | eqid 2731 | . . . . . . 7 β’ (π βΎs (π β (Baseβπ ))) = (π βΎs (π β (Baseβπ ))) | |
17 | rimrhm 20388 | . . . . . . . 8 β’ (π β (π RingIso π) β π β (π RingHom π)) | |
18 | 17 | adantr 480 | . . . . . . 7 β’ ((π β (π RingIso π) β§ π β CRing) β π β (π RingHom π)) |
19 | simpr 484 | . . . . . . 7 β’ ((π β (π RingIso π) β§ π β CRing) β π β CRing) | |
20 | 19 | crngringd 20141 | . . . . . . . 8 β’ ((π β (π RingIso π) β§ π β CRing) β π β Ring) |
21 | 4 | subrgid 20464 | . . . . . . . 8 β’ (π β Ring β (Baseβπ ) β (SubRingβπ )) |
22 | 20, 21 | syl 17 | . . . . . . 7 β’ ((π β (π RingIso π) β§ π β CRing) β (Baseβπ ) β (SubRingβπ )) |
23 | 16, 18, 19, 22 | imacrhmcl 41394 | . . . . . 6 β’ ((π β (π RingIso π) β§ π β CRing) β (π βΎs (π β (Baseβπ ))) β CRing) |
24 | 15, 23 | eqeltrd 2832 | . . . . 5 β’ ((π β (π RingIso π) β§ π β CRing) β π β CRing) |
25 | 24 | ex 412 | . . . 4 β’ (π β (π RingIso π) β (π β CRing β π β CRing)) |
26 | 25 | exlimiv 1932 | . . 3 β’ (βπ π β (π RingIso π) β (π β CRing β π β CRing)) |
27 | 26 | imp 406 | . 2 β’ ((βπ π β (π RingIso π) β§ π β CRing) β π β CRing) |
28 | 3, 27 | sylanb 580 | 1 β’ ((π βπ π β§ π β CRing) β π β CRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 βwex 1780 β wcel 2105 β wne 2939 β c0 4322 class class class wbr 5148 β cima 5679 βontoβwfo 6541 β1-1-ontoβwf1o 6542 βcfv 6543 (class class class)co 7412 Basecbs 17149 βΎs cress 17178 Ringcrg 20128 CRingccrg 20129 RingHom crh 20361 RingIso crs 20362 βπ cric 20363 SubRingcsubrg 20458 |
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 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-rmo 3375 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 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 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-ress 17179 df-plusg 17215 df-mulr 17216 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-subg 19040 df-ghm 19129 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-rhm 20364 df-rim 20365 df-ric 20367 df-subrng 20435 df-subrg 20460 |
This theorem is referenced by: riccrng 41402 |
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