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Mirrors > Home > MPE Home > Th. List > cntzidss | Structured version Visualization version GIF version |
Description: If the elements of 𝑆 commute, the elements of a subset 𝑇 also commute. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
cntzmhm.z | ⊢ 𝑍 = (Cntz‘𝐺) |
Ref | Expression |
---|---|
cntzidss | ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ (𝑍‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . 2 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆) | |
2 | simpl 486 | . . 3 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ (𝑍‘𝑆)) | |
3 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | cntzmhm.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺) | |
5 | 3, 4 | cntzssv 18450 | . . . . 5 ⊢ (𝑍‘𝑆) ⊆ (Base‘𝐺) |
6 | 2, 5 | sstrdi 3927 | . . . 4 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝐺)) |
7 | 3, 4 | cntz2ss 18455 | . . . 4 ⊢ ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇)) |
8 | 6, 7 | sylancom 591 | . . 3 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇)) |
9 | 2, 8 | sstrd 3925 | . 2 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ (𝑍‘𝑇)) |
10 | 1, 9 | sstrd 3925 | 1 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ (𝑍‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ⊆ wss 3881 ‘cfv 6324 Basecbs 16475 Cntzccntz 18437 |
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 |
This theorem is referenced by: gsumzres 19022 gsumzf1o 19025 gsumzaddlem 19034 gsumzadd 19035 gsumzsplit 19040 gsumconst 19047 gsumpt 19075 dprdfadd 19135 |
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