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Mirrors > Home > MPE Home > Th. List > cntzcmnf | Structured version Visualization version GIF version |
Description: Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.) |
Ref | Expression |
---|---|
cntzcmnf.b | β’ π΅ = (BaseβπΊ) |
cntzcmnf.z | β’ π = (CntzβπΊ) |
cntzcmnf.g | β’ (π β πΊ β CMnd) |
cntzcmnf.f | β’ (π β πΉ:π΄βΆπ΅) |
Ref | Expression |
---|---|
cntzcmnf | β’ (π β ran πΉ β (πβran πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzcmnf.f | . . 3 β’ (π β πΉ:π΄βΆπ΅) | |
2 | 1 | frnd 6719 | . 2 β’ (π β ran πΉ β π΅) |
3 | cntzcmnf.g | . . 3 β’ (π β πΊ β CMnd) | |
4 | cntzcmnf.b | . . . 4 β’ π΅ = (BaseβπΊ) | |
5 | cntzcmnf.z | . . . 4 β’ π = (CntzβπΊ) | |
6 | 4, 5 | cntzcmn 19760 | . . 3 β’ ((πΊ β CMnd β§ ran πΉ β π΅) β (πβran πΉ) = π΅) |
7 | 3, 2, 6 | syl2anc 583 | . 2 β’ (π β (πβran πΉ) = π΅) |
8 | 2, 7 | sseqtrrd 4018 | 1 β’ (π β ran πΉ β (πβran πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3943 ran crn 5670 βΆwf 6533 βcfv 6537 Basecbs 17153 Cntzccntz 19231 CMndccmn 19700 |
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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-ral 3056 df-rex 3065 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-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 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-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-cntz 19233 df-cmn 19702 |
This theorem is referenced by: gsumres 19833 gsumcl2 19834 gsumf1o 19836 gsumsubmcl 19839 gsumsplit 19848 gsummhm 19858 gsumfsum 21328 wilthlem3 26957 |
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