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Mirrors > Home > MPE Home > Th. List > lsmcntzr | Structured version Visualization version GIF version |
Description: The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
lsmcntz.p | ⊢ ⊕ = (LSSum‘𝐺) |
lsmcntz.s | ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) |
lsmcntz.t | ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
lsmcntz.u | ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) |
lsmcntz.z | ⊢ 𝑍 = (Cntz‘𝐺) |
Ref | Expression |
---|---|
lsmcntzr | ⊢ (𝜑 → (𝑆 ⊆ (𝑍‘(𝑇 ⊕ 𝑈)) ↔ (𝑆 ⊆ (𝑍‘𝑇) ∧ 𝑆 ⊆ (𝑍‘𝑈)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmcntz.p | . . 3 ⊢ ⊕ = (LSSum‘𝐺) | |
2 | lsmcntz.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
3 | lsmcntz.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
4 | lsmcntz.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
5 | lsmcntz.z | . . 3 ⊢ 𝑍 = (Cntz‘𝐺) | |
6 | 1, 2, 3, 4, 5 | lsmcntz 17915 | . 2 ⊢ (𝜑 → ((𝑇 ⊕ 𝑈) ⊆ (𝑍‘𝑆) ↔ (𝑇 ⊆ (𝑍‘𝑆) ∧ 𝑈 ⊆ (𝑍‘𝑆)))) |
7 | subgrcl 17422 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
8 | grpmnd 17252 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
9 | 4, 7, 8 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
10 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | 10 | subgss 17418 | . . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
12 | 2, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺)) |
13 | 10 | subgss 17418 | . . . . 5 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
14 | 3, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺)) |
15 | 10, 1 | lsmssv 17881 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺)) |
16 | 9, 12, 14, 15 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺)) |
17 | 10 | subgss 17418 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
18 | 4, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐺)) |
19 | 10, 5 | cntzrec 17589 | . . 3 ⊢ (((𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → ((𝑇 ⊕ 𝑈) ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘(𝑇 ⊕ 𝑈)))) |
20 | 16, 18, 19 | syl2anc 691 | . 2 ⊢ (𝜑 → ((𝑇 ⊕ 𝑈) ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘(𝑇 ⊕ 𝑈)))) |
21 | 10, 5 | cntzrec 17589 | . . . 4 ⊢ ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑇 ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘𝑇))) |
22 | 12, 18, 21 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑇 ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘𝑇))) |
23 | 10, 5 | cntzrec 17589 | . . . 4 ⊢ ((𝑈 ⊆ (Base‘𝐺) ∧ 𝑆 ⊆ (Base‘𝐺)) → (𝑈 ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘𝑈))) |
24 | 14, 18, 23 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑈 ⊆ (𝑍‘𝑆) ↔ 𝑆 ⊆ (𝑍‘𝑈))) |
25 | 22, 24 | anbi12d 743 | . 2 ⊢ (𝜑 → ((𝑇 ⊆ (𝑍‘𝑆) ∧ 𝑈 ⊆ (𝑍‘𝑆)) ↔ (𝑆 ⊆ (𝑍‘𝑇) ∧ 𝑆 ⊆ (𝑍‘𝑈)))) |
26 | 6, 20, 25 | 3bitr3d 297 | 1 ⊢ (𝜑 → (𝑆 ⊆ (𝑍‘(𝑇 ⊕ 𝑈)) ↔ (𝑆 ⊆ (𝑍‘𝑇) ∧ 𝑆 ⊆ (𝑍‘𝑈)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Mndcmnd 17117 Grpcgrp 17245 SubGrpcsubg 17411 Cntzccntz 17571 LSSumclsm 17872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-subg 17414 df-cntz 17573 df-lsm 17874 |
This theorem is referenced by: (None) |
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