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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 6494 | . 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 18953 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ ran 𝐹 ⊆ 𝐵) → (𝑍‘ran 𝐹) = 𝐵) |
7 | 3, 2, 6 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑍‘ran 𝐹) = 𝐵) |
8 | 2, 7 | sseqtrrd 3956 | 1 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ran crn 5520 ⟶wf 6320 ‘cfv 6324 Basecbs 16475 Cntzccntz 18437 CMndccmn 18898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-cntz 18439 df-cmn 18900 |
This theorem is referenced by: gsumres 19026 gsumcl2 19027 gsumf1o 19029 gsumsubmcl 19032 gsumsplit 19041 gsummhm 19051 gsumfsum 20158 wilthlem3 25655 |
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