![]() |
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 2729 | . . . 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 21386 | . . . . 5 β’ β€ = (Baseββ€ring) | |
3 | 2 | a1i 11 | . . . 4 β’ (π β β0 β β€ = (Baseββ€ring)) |
4 | eqid 2728 | . . . 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 7461 | . . . 4 β’ (π β β0 β (β€ring ~QG ((RSpanββ€ring)β{π})) β V) | |
6 | zringring 21382 | . . . . 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 17532 | . . 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 2728 | . . . . . 6 β’ (RSpanββ€ring) = (RSpanββ€ring) | |
10 | znzrhfo.y | . . . . . 6 β’ π = (β€/nβ€βπ) | |
11 | eqid 2728 | . . . . . 6 β’ (β€ring ~QG ((RSpanββ€ring)β{π})) = (β€ring ~QG ((RSpanββ€ring)β{π})) | |
12 | 9, 10, 11 | znbas 21484 | . . . . 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 2786 | . . . 4 β’ (π β β0 β (β€ / (β€ring ~QG ((RSpanββ€ring)β{π}))) = π΅) |
15 | foeq3 6814 | . . . 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 21486 | . . 3 β’ (π β β0 β πΏ = (π₯ β β€ β¦ [π₯](β€ring ~QG ((RSpanββ€ring)β{π})))) |
20 | foeq1 6812 | . . 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 256 | 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 3473 {csn 4632 β¦ cmpt 5235 βontoβwfo 6551 βcfv 6553 (class class class)co 7426 [cec 8729 / cqs 8730 β0cn0 12510 β€cz 12596 Basecbs 17187 /s cqus 17494 ~QG cqg 19084 Ringcrg 20180 RSpancrsp 21110 β€ringczring 21379 β€RHomczrh 21432 β€/nβ€czn 21435 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 ax-mulf 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 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 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-ec 8733 df-qs 8737 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-seq 14007 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-imas 17497 df-qus 17498 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-nsg 19086 df-eqg 19087 df-ghm 19175 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-lsp 20863 df-sra 21065 df-rgmod 21066 df-lidl 21111 df-rsp 21112 df-2idl 21151 df-cnfld 21287 df-zring 21380 df-zrh 21436 df-zn 21439 |
This theorem is referenced by: zncyg 21489 znf1o 21492 zzngim 21493 znfld 21501 znunit 21504 znrrg 21506 cygznlem2a 21508 cygznlem3 21510 dchrelbas4 27196 dchrzrhcl 27198 lgsdchrval 27307 lgsdchr 27308 rpvmasumlem 27440 dchrmusum2 27447 dchrvmasumlem3 27452 dchrisum0ff 27460 dchrisum0flblem1 27461 rpvmasum2 27465 dchrisum0re 27466 dchrisum0lem2a 27470 dirith 27482 znfermltl 33102 |
Copyright terms: Public domain | W3C validator |