![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > znzrhfo | Structured version Visualization version GIF version |
Description: The β€ ring homomorphism is a surjection onto β€ / πβ€. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
znzrhfo.y | β’ π = (β€/nβ€βπ) |
znzrhfo.b | β’ π΅ = (Baseβπ) |
znzrhfo.2 | β’ πΏ = (β€RHomβπ) |
Ref | Expression |
---|---|
znzrhfo | β’ (π β β0 β πΏ:β€βontoβπ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2727 | . . . 4 β’ (π β β0 β (β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))) = (β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) | |
2 | zringbas 21335 | . . . . 5 β’ β€ = (Baseββ€ring) | |
3 | 2 | a1i 11 | . . . 4 β’ (π β β0 β β€ = (Baseββ€ring)) |
4 | eqid 2726 | . . . 4 β’ (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))) = (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))) | |
5 | ovexd 7439 | . . . 4 β’ (π β β0 β (β€ring ~QG ((RSpanββ€ring)β{π})) β V) | |
6 | zringring 21331 | . . . . 5 β’ β€ring β Ring | |
7 | 6 | a1i 11 | . . . 4 β’ (π β β0 β β€ring β Ring) |
8 | 1, 3, 4, 5, 7 | quslem 17495 | . . 3 β’ (π β β0 β (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))):β€βontoβ(β€ / (β€ring ~QG ((RSpanββ€ring)β{π})))) |
9 | eqid 2726 | . . . . . 6 β’ (RSpanββ€ring) = (RSpanββ€ring) | |
10 | znzrhfo.y | . . . . . 6 β’ π = (β€/nβ€βπ) | |
11 | eqid 2726 | . . . . . 6 β’ (β€ring ~QG ((RSpanββ€ring)β{π})) = (β€ring ~QG ((RSpanββ€ring)β{π})) | |
12 | 9, 10, 11 | znbas 21433 | . . . . 5 β’ (π β β0 β (β€ / (β€ring ~QG ((RSpanββ€ring)β{π}))) = (Baseβπ)) |
13 | znzrhfo.b | . . . . 5 β’ π΅ = (Baseβπ) | |
14 | 12, 13 | eqtr4di 2784 | . . . 4 β’ (π β β0 β (β€ / (β€ring ~QG ((RSpanββ€ring)β{π}))) = π΅) |
15 | foeq3 6796 | . . . 4 β’ ((β€ / (β€ring ~QG ((RSpanββ€ring)β{π}))) = π΅ β ((π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))):β€βontoβ(β€ / (β€ring ~QG ((RSpanββ€ring)β{π}))) β (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))):β€βontoβπ΅)) | |
16 | 14, 15 | syl 17 | . . 3 β’ (π β β0 β ((π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))):β€βontoβ(β€ / (β€ring ~QG ((RSpanββ€ring)β{π}))) β (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))):β€βontoβπ΅)) |
17 | 8, 16 | mpbid 231 | . 2 β’ (π β β0 β (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))):β€βontoβπ΅) |
18 | znzrhfo.2 | . . . 4 β’ πΏ = (β€RHomβπ) | |
19 | 9, 11, 10, 18 | znzrh2 21435 | . . 3 β’ (π β β0 β πΏ = (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π})))) |
20 | foeq1 6794 | . . 3 β’ (πΏ = (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))) β (πΏ:β€βontoβπ΅ β (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))):β€βontoβπ΅)) | |
21 | 19, 20 | syl 17 | . 2 β’ (π β β0 β (πΏ:β€βontoβπ΅ β (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π}))):β€βontoβπ΅)) |
22 | 17, 21 | mpbird 257 | 1 β’ (π β β0 β πΏ:β€βontoβπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 Vcvv 3468 {csn 4623 β¦ cmpt 5224 βontoβwfo 6534 βcfv 6536 (class class class)co 7404 [cec 8700 / cqs 8701 β0cn0 12473 β€cz 12559 Basecbs 17150 /s cqus 17457 ~QG cqg 19046 Ringcrg 20135 RSpancrsp 21063 β€ringczring 21328 β€RHomczrh 21381 β€/nβ€czn 21384 |
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-int 4944 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-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-ec 8704 df-qs 8708 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 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-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-0g 17393 df-imas 17460 df-qus 17461 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-nsg 19048 df-eqg 19049 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-oppr 20233 df-rhm 20371 df-subrng 20443 df-subrg 20468 df-lmod 20705 df-lss 20776 df-lsp 20816 df-sra 21018 df-rgmod 21019 df-lidl 21064 df-rsp 21065 df-2idl 21104 df-cnfld 21236 df-zring 21329 df-zrh 21385 df-zn 21388 |
This theorem is referenced by: zncyg 21438 znf1o 21441 zzngim 21442 znfld 21450 znunit 21453 znrrg 21455 cygznlem2a 21457 cygznlem3 21459 dchrelbas4 27126 dchrzrhcl 27128 lgsdchrval 27237 lgsdchr 27238 rpvmasumlem 27370 dchrmusum2 27377 dchrvmasumlem3 27382 dchrisum0ff 27390 dchrisum0flblem1 27391 rpvmasum2 27395 dchrisum0re 27396 dchrisum0lem2a 27400 dirith 27412 znfermltl 32984 |
Copyright terms: Public domain | W3C validator |