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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvexchlem2 | Structured version Visualization version GIF version |
Description: Lemma for lcvexch 38435. (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 20824 | . . . . 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 19614 | . . 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 1371 | . 2 ⊢ (𝜑 → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = ((𝑅 ⊕ 𝑇) ∩ 𝑈)) |
15 | 2 | lssincl 20831 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆) |
16 | 1, 7, 9, 15 | syl3anc 1369 | . . . 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 19606 | . . 3 ⊢ ((𝑅 ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ⊆ 𝑅) → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = 𝑅) |
20 | 6, 17, 18, 19 | syl3anc 1369 | . 2 ⊢ (𝜑 → (𝑅 ⊕ (𝑇 ∩ 𝑈)) = 𝑅) |
21 | 14, 20 | eqtr3d 2769 | 1 ⊢ (𝜑 → ((𝑅 ⊕ 𝑇) ∩ 𝑈) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∩ cin 3943 ⊆ wss 3944 ‘cfv 6542 (class class class)co 7414 SubGrpcsubg 19059 LSSumclsm 19573 LModclmod 20725 LSubSpclss 20797 ⋖L clcv 38414 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-0g 17408 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19062 df-lsm 19575 df-mgp 20059 df-ur 20106 df-ring 20159 df-lmod 20727 df-lss 20798 |
This theorem is referenced by: lcvexchlem4 38433 |
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