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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 485 | . 2 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆) | |
2 | simpl 483 | . . 3 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ (𝑍‘𝑆)) | |
3 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | cntzmhm.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺) | |
5 | 3, 4 | cntzssv 18396 | . . . . 5 ⊢ (𝑍‘𝑆) ⊆ (Base‘𝐺) |
6 | 2, 5 | sstrdi 3976 | . . . 4 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝐺)) |
7 | 3, 4 | cntz2ss 18401 | . . . 4 ⊢ ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇)) |
8 | 6, 7 | sylancom 588 | . . 3 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇)) |
9 | 2, 8 | sstrd 3974 | . 2 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ (𝑍‘𝑇)) |
10 | 1, 9 | sstrd 3974 | 1 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ (𝑍‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ⊆ wss 3933 ‘cfv 6348 Basecbs 16471 Cntzccntz 18383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-cntz 18385 |
This theorem is referenced by: gsumzres 18958 gsumzf1o 18961 gsumzaddlem 18970 gsumzadd 18971 gsumzsplit 18976 gsumconst 18983 gsumpt 19011 dprdfadd 19071 |
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