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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvexchlem3 | Structured version Visualization version GIF version |
Description: Lemma for lcvexch 36167. (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lcvexch.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lcvexch.p | ⊢ ⊕ = (LSSum‘𝑊) |
lcvexch.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lcvexch.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lcvexch.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
lcvexch.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lcvexch.q | ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
lcvexch.d | ⊢ (𝜑 → 𝑇 ⊆ 𝑅) |
lcvexch.e | ⊢ (𝜑 → 𝑅 ⊆ (𝑇 ⊕ 𝑈)) |
Ref | Expression |
---|---|
lcvexchlem3 | ⊢ (𝜑 → ((𝑅 ∩ 𝑈) ⊕ 𝑇) = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcvexch.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lcvexch.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 2 | lsssssubg 19722 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
5 | lcvexch.q | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) | |
6 | 4, 5 | sseldd 3966 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
7 | lcvexch.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 4, 7 | sseldd 3966 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
9 | lcvexch.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
10 | 4, 9 | sseldd 3966 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
11 | lcvexch.d | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑅) | |
12 | lcvexch.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
13 | 12 | lsmmod2 18794 | . . 3 ⊢ (((𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊)) ∧ 𝑇 ⊆ 𝑅) → (𝑅 ∩ (𝑈 ⊕ 𝑇)) = ((𝑅 ∩ 𝑈) ⊕ 𝑇)) |
14 | 6, 8, 10, 11, 13 | syl31anc 1368 | . 2 ⊢ (𝜑 → (𝑅 ∩ (𝑈 ⊕ 𝑇)) = ((𝑅 ∩ 𝑈) ⊕ 𝑇)) |
15 | lcvexch.e | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ (𝑇 ⊕ 𝑈)) | |
16 | lmodabl 19673 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
17 | 1, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
18 | 12 | lsmcom 18970 | . . . . 5 ⊢ ((𝑊 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
19 | 17, 10, 8, 18 | syl3anc 1366 | . . . 4 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇)) |
20 | 15, 19 | sseqtrd 4005 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑇)) |
21 | df-ss 3950 | . . 3 ⊢ (𝑅 ⊆ (𝑈 ⊕ 𝑇) ↔ (𝑅 ∩ (𝑈 ⊕ 𝑇)) = 𝑅) | |
22 | 20, 21 | sylib 220 | . 2 ⊢ (𝜑 → (𝑅 ∩ (𝑈 ⊕ 𝑇)) = 𝑅) |
23 | 14, 22 | eqtr3d 2856 | 1 ⊢ (𝜑 → ((𝑅 ∩ 𝑈) ⊕ 𝑇) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ∩ cin 3933 ⊆ wss 3934 ‘cfv 6348 (class class class)co 7148 SubGrpcsubg 18265 LSSumclsm 18751 Abelcabl 18899 LModclmod 19626 LSubSpclss 19695 ⋖L clcv 36146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-lmod 19628 df-lss 19696 |
This theorem is referenced by: lcvexchlem5 36166 |
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