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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 483 | . 2 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆) | |
2 | simpl 481 | . . 3 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ (𝑍‘𝑆)) | |
3 | eqid 2725 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | cntzmhm.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺) | |
5 | 3, 4 | cntzssv 19308 | . . . . 5 ⊢ (𝑍‘𝑆) ⊆ (Base‘𝐺) |
6 | 2, 5 | sstrdi 3989 | . . . 4 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝐺)) |
7 | 3, 4 | cntz2ss 19315 | . . . 4 ⊢ ((𝑆 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇)) |
8 | 6, 7 | sylancom 586 | . . 3 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇)) |
9 | 2, 8 | sstrd 3987 | . 2 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ (𝑍‘𝑇)) |
10 | 1, 9 | sstrd 3987 | 1 ⊢ ((𝑆 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ (𝑍‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ⊆ wss 3944 ‘cfv 6549 Basecbs 17199 Cntzccntz 19295 |
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 19297 |
This theorem is referenced by: gsumzres 19893 gsumzf1o 19896 gsumzaddlem 19905 gsumzadd 19906 gsumzsplit 19911 gsumconst 19918 gsumpt 19946 dprdfadd 20006 |
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