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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvexchlem2 | Structured version Visualization version GIF version |
Description: Lemma for lcvexch 38563. (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 20841 | . . . . 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 3974 | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
7 | lcvexch.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
8 | 4, 7 | sseldd 3974 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
9 | lcvexch.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
10 | 4, 9 | sseldd 3974 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
11 | lcvexch.b | . . 3 ⊢ (𝜑 → 𝑅 ⊆ 𝑈) | |
12 | lcvexch.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
13 | 12 | lsmmod 19629 | . . 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 20848 | . . . . 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 3974 | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊)) |
18 | lcvexch.a | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑅) | |
19 | 12 | lsmss2 19621 | . . 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 3940 ⊆ wss 3941 ‘cfv 6543 (class class class)co 7413 SubGrpcsubg 19074 LSSumclsm 19588 LModclmod 20742 LSubSpclss 20814 ⋖L clcv 38542 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-0g 17417 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-lsm 19590 df-mgp 20074 df-ur 20121 df-ring 20174 df-lmod 20744 df-lss 20815 |
This theorem is referenced by: lcvexchlem4 38561 |
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