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Mirrors > Home > MPE Home > Th. List > lsmdisj2a | Structured version Visualization version GIF version |
Description: Association of the disjointness constraint in 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)) |
lsmdisj.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
lsmdisj2a | ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmcntz.p | . . . 4 ⊢ ⊕ = (LSSum‘𝐺) | |
2 | lsmcntz.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) | |
3 | 2 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → 𝑆 ∈ (SubGrp‘𝐺)) |
4 | lsmcntz.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
5 | 4 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → 𝑇 ∈ (SubGrp‘𝐺)) |
6 | lsmcntz.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
7 | 6 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → 𝑈 ∈ (SubGrp‘𝐺)) |
8 | lsmdisj.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
9 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) | |
10 | simprr 772 | . . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → (𝑆 ∩ 𝑇) = { 0 }) | |
11 | 1, 3, 5, 7, 8, 9, 10 | lsmdisj2 19534 | . . 3 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 }) |
12 | 1, 3, 5, 7, 8, 9 | lsmdisj 19533 | . . . 4 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 })) |
13 | 12 | simpld 496 | . . 3 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → (𝑆 ∩ 𝑈) = { 0 }) |
14 | 11, 13 | jca 513 | . 2 ⊢ ((𝜑 ∧ (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) → ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) |
15 | incom 4199 | . . . 4 ⊢ ((𝑆 ⊕ 𝑇) ∩ 𝑈) = (𝑈 ∩ (𝑆 ⊕ 𝑇)) | |
16 | 2 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → 𝑆 ∈ (SubGrp‘𝐺)) |
17 | 6 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → 𝑈 ∈ (SubGrp‘𝐺)) |
18 | 4 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → 𝑇 ∈ (SubGrp‘𝐺)) |
19 | incom 4199 | . . . . . 6 ⊢ ((𝑆 ⊕ 𝑈) ∩ 𝑇) = (𝑇 ∩ (𝑆 ⊕ 𝑈)) | |
20 | simprl 770 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 }) | |
21 | 19, 20 | eqtrid 2785 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 }) |
22 | simprr 772 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑆 ∩ 𝑈) = { 0 }) | |
23 | 1, 16, 17, 18, 8, 21, 22 | lsmdisj2 19534 | . . . 4 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑈 ∩ (𝑆 ⊕ 𝑇)) = { 0 }) |
24 | 15, 23 | eqtrid 2785 | . . 3 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) |
25 | incom 4199 | . . . 4 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆) | |
26 | 1, 18, 16, 17, 8, 20 | lsmdisjr 19536 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → ((𝑇 ∩ 𝑆) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 })) |
27 | 26 | simpld 496 | . . . 4 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑇 ∩ 𝑆) = { 0 }) |
28 | 25, 27 | eqtrid 2785 | . . 3 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (𝑆 ∩ 𝑇) = { 0 }) |
29 | 24, 28 | jca 513 | . 2 ⊢ ((𝜑 ∧ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) → (((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 })) |
30 | 14, 29 | impbida 800 | 1 ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3945 {csn 4624 ‘cfv 6535 (class class class)co 7396 0gc0g 17372 SubGrpcsubg 18985 LSSumclsm 19486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-0g 17374 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-grp 18809 df-minusg 18810 df-subg 18988 df-lsm 19488 |
This theorem is referenced by: lsmdisj3a 19541 |
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