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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 | ⊢ 𝐵 = (Base‘𝐺) |
cntzcmnf.z | ⊢ 𝑍 = (Cntz‘𝐺) |
cntzcmnf.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
cntzcmnf.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
cntzcmnf | ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzcmnf.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | frnd 6731 | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) |
3 | cntzcmnf.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | cntzcmnf.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | cntzcmnf.z | . . . 4 ⊢ 𝑍 = (Cntz‘𝐺) | |
6 | 4, 5 | cntzcmn 19807 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ ran 𝐹 ⊆ 𝐵) → (𝑍‘ran 𝐹) = 𝐵) |
7 | 3, 2, 6 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝑍‘ran 𝐹) = 𝐵) |
8 | 2, 7 | sseqtrrd 4018 | 1 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ran crn 5679 ⟶wf 6545 ‘cfv 6549 Basecbs 17183 Cntzccntz 19278 CMndccmn 19747 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-cntz 19280 df-cmn 19749 |
This theorem is referenced by: gsumres 19880 gsumcl2 19881 gsumf1o 19883 gsumsubmcl 19886 gsumsplit 19895 gsummhm 19905 gsumfsum 21384 wilthlem3 27047 |
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