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Mirrors > Home > MPE Home > Th. List > zrhval2 | Structured version Visualization version GIF version |
Description: Alternate value of the โคRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
zrhval.l | โข ๐ฟ = (โคRHomโ๐ ) |
zrhval2.m | โข ยท = (.gโ๐ ) |
zrhval2.1 | โข 1 = (1rโ๐ ) |
Ref | Expression |
---|---|
zrhval2 | โข (๐ โ Ring โ ๐ฟ = (๐ โ โค โฆ (๐ ยท 1 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrhval.l | . . 3 โข ๐ฟ = (โคRHomโ๐ ) | |
2 | 1 | zrhval 21389 | . 2 โข ๐ฟ = โช (โคring RingHom ๐ ) |
3 | zrhval2.m | . . . . 5 โข ยท = (.gโ๐ ) | |
4 | eqid 2726 | . . . . 5 โข (๐ โ โค โฆ (๐ ยท 1 )) = (๐ โ โค โฆ (๐ ยท 1 )) | |
5 | zrhval2.1 | . . . . 5 โข 1 = (1rโ๐ ) | |
6 | 3, 4, 5 | mulgrhm2 21360 | . . . 4 โข (๐ โ Ring โ (โคring RingHom ๐ ) = {(๐ โ โค โฆ (๐ ยท 1 ))}) |
7 | 6 | unieqd 4915 | . . 3 โข (๐ โ Ring โ โช (โคring RingHom ๐ ) = โช {(๐ โ โค โฆ (๐ ยท 1 ))}) |
8 | zex 12568 | . . . . 5 โข โค โ V | |
9 | 8 | mptex 7219 | . . . 4 โข (๐ โ โค โฆ (๐ ยท 1 )) โ V |
10 | 9 | unisn 4923 | . . 3 โข โช {(๐ โ โค โฆ (๐ ยท 1 ))} = (๐ โ โค โฆ (๐ ยท 1 )) |
11 | 7, 10 | eqtrdi 2782 | . 2 โข (๐ โ Ring โ โช (โคring RingHom ๐ ) = (๐ โ โค โฆ (๐ ยท 1 ))) |
12 | 2, 11 | eqtrid 2778 | 1 โข (๐ โ Ring โ ๐ฟ = (๐ โ โค โฆ (๐ ยท 1 ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 {csn 4623 โช cuni 4902 โฆ cmpt 5224 โcfv 6536 (class class class)co 7404 โคcz 12559 .gcmg 18992 1rcur 20083 Ringcrg 20135 RingHom crh 20368 โคringczring 21328 โคRHomczrh 21381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-seq 13970 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-grp 18863 df-minusg 18864 df-mulg 18993 df-subg 19047 df-ghm 19136 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-rhm 20371 df-subrng 20443 df-subrg 20468 df-cnfld 21236 df-zring 21329 df-zrh 21385 |
This theorem is referenced by: zrhmulg 21391 zrhrhmb 21392 zncyg 21438 zrhchr 33485 zrhre 33528 |
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