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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 6735 | . 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 19802 | . . 3 β’ ((πΊ β CMnd β§ ran πΉ β π΅) β (πβran πΉ) = π΅) |
7 | 3, 2, 6 | syl2anc 582 | . 2 β’ (π β (πβran πΉ) = π΅) |
8 | 2, 7 | sseqtrrd 4023 | 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 3949 ran crn 5683 βΆwf 6549 βcfv 6553 Basecbs 17187 Cntzccntz 19273 CMndccmn 19742 |
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 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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-ral 3059 df-rex 3068 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-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 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-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-cntz 19275 df-cmn 19744 |
This theorem is referenced by: gsumres 19875 gsumcl2 19876 gsumf1o 19878 gsumsubmcl 19881 gsumsplit 19890 gsummhm 19900 gsumfsum 21374 wilthlem3 27022 |
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