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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvexchlem2 | Structured version Visualization version GIF version |
Description: Lemma for lcvexch 35649. (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 19464 | . . . . 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 3853 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
7 | lcvexch.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
8 | 4, 7 | sseldd 3853 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
9 | lcvexch.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
10 | 4, 9 | sseldd 3853 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
11 | lcvexch.b | . . 3 ⊢ (𝜑 → 𝑅 ⊆ 𝑈) | |
12 | lcvexch.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
13 | 12 | lsmmod 18571 | . . 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 1353 | . 2 ⊢ (𝜑 → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = ((𝑅 ⊕ 𝑇) ∩ 𝑈)) |
15 | 2 | lssincl 19471 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆) |
16 | 1, 7, 9, 15 | syl3anc 1351 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ 𝑆) |
17 | 4, 16 | sseldd 3853 | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊)) |
18 | lcvexch.a | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑅) | |
19 | 12 | lsmss2 18564 | . . 3 ⊢ ((𝑅 ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ⊆ 𝑅) → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = 𝑅) |
20 | 6, 17, 18, 19 | syl3anc 1351 | . 2 ⊢ (𝜑 → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = 𝑅) |
21 | 14, 20 | eqtr3d 2810 | 1 ⊢ (𝜑 → ((𝑅 ⊕ 𝑇) ∩ 𝑈) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ∩ cin 3822 ⊆ wss 3823 ‘cfv 6185 (class class class)co 6974 SubGrpcsubg 18069 LSSumclsm 18532 LModclmod 19368 LSubSpclss 19437 ⋖L clcv 35628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-0g 16569 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-subg 18072 df-lsm 18534 df-mgp 18975 df-ur 18987 df-ring 19034 df-lmod 19370 df-lss 19438 |
This theorem is referenced by: lcvexchlem4 35647 |
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