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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvexchlem2 | Structured version Visualization version GIF version |
Description: Lemma for lcvexch 38685. (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.r | ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
lcvexch.a | ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑅) |
lcvexch.b | ⊢ (𝜑 → 𝑅 ⊆ 𝑈) |
Ref | Expression |
---|---|
lcvexchlem2 | ⊢ (𝜑 → ((𝑅 ⊕ 𝑇) ∩ 𝑈) = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcvexch.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lcvexch.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 2 | lsssssubg 20882 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
5 | lcvexch.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) | |
6 | 4, 5 | sseldd 3979 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
7 | lcvexch.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
8 | 4, 7 | sseldd 3979 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
9 | lcvexch.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
10 | 4, 9 | sseldd 3979 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
11 | lcvexch.b | . . 3 ⊢ (𝜑 → 𝑅 ⊆ 𝑈) | |
12 | lcvexch.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
13 | 12 | lsmmod 19668 | . . 3 ⊢ (((𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) ∧ 𝑅 ⊆ 𝑈) → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = ((𝑅 ⊕ 𝑇) ∩ 𝑈)) |
14 | 6, 8, 10, 11, 13 | syl31anc 1370 | . 2 ⊢ (𝜑 → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = ((𝑅 ⊕ 𝑇) ∩ 𝑈)) |
15 | 2 | lssincl 20889 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆) |
16 | 1, 7, 9, 15 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ 𝑆) |
17 | 4, 16 | sseldd 3979 | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊)) |
18 | lcvexch.a | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑅) | |
19 | 12 | lsmss2 19660 | . . 3 ⊢ ((𝑅 ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ⊆ 𝑅) → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = 𝑅) |
20 | 6, 17, 18, 19 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = 𝑅) |
21 | 14, 20 | eqtr3d 2767 | 1 ⊢ (𝜑 → ((𝑅 ⊕ 𝑇) ∩ 𝑈) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3945 ⊆ wss 3946 ‘cfv 6553 (class class class)co 7423 SubGrpcsubg 19109 LSSumclsm 19627 LModclmod 20783 LSubSpclss 20855 ⋖L clcv 38664 |
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 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-0g 17451 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19112 df-lsm 19629 df-mgp 20113 df-ur 20160 df-ring 20213 df-lmod 20785 df-lss 20856 |
This theorem is referenced by: lcvexchlem4 38683 |
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