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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvexchlem1 | Structured version Visualization version GIF version |
Description: Lemma for lcvexch 38403. (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) |
Ref | Expression |
---|---|
lcvexchlem1 | ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcvexch.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lcvexch.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 2 | lsssssubg 20797 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
4 | 1, 3 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
5 | lcvexch.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
6 | 4, 5 | sseldd 3976 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
7 | lcvexch.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 4, 7 | sseldd 3976 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
9 | lcvexch.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
10 | 9 | lsmub1 19569 | . . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) |
11 | 6, 8, 10 | syl2anc 583 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) |
12 | inss2 4222 | . . . . 5 ⊢ (𝑇 ∩ 𝑈) ⊆ 𝑈 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑈) |
14 | 11, 13 | 2thd 265 | . . 3 ⊢ (𝜑 → (𝑇 ⊆ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊆ 𝑈)) |
15 | 9 | lsmss2b 19580 | . . . . . . 7 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) |
16 | 6, 8, 15 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) |
17 | eqcom 2731 | . . . . . 6 ⊢ ((𝑇 ⊕ 𝑈) = 𝑇 ↔ 𝑇 = (𝑇 ⊕ 𝑈)) | |
18 | 16, 17 | bitrdi 287 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊆ 𝑇 ↔ 𝑇 = (𝑇 ⊕ 𝑈))) |
19 | sseqin2 4208 | . . . . 5 ⊢ (𝑈 ⊆ 𝑇 ↔ (𝑇 ∩ 𝑈) = 𝑈) | |
20 | 18, 19 | bitr3di 286 | . . . 4 ⊢ (𝜑 → (𝑇 = (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) = 𝑈)) |
21 | 20 | necon3bid 2977 | . . 3 ⊢ (𝜑 → (𝑇 ≠ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ≠ 𝑈)) |
22 | 14, 21 | anbi12d 630 | . 2 ⊢ (𝜑 → ((𝑇 ⊆ (𝑇 ⊕ 𝑈) ∧ 𝑇 ≠ (𝑇 ⊕ 𝑈)) ↔ ((𝑇 ∩ 𝑈) ⊆ 𝑈 ∧ (𝑇 ∩ 𝑈) ≠ 𝑈))) |
23 | df-pss 3960 | . 2 ⊢ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ⊆ (𝑇 ⊕ 𝑈) ∧ 𝑇 ≠ (𝑇 ⊕ 𝑈))) | |
24 | df-pss 3960 | . 2 ⊢ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ↔ ((𝑇 ∩ 𝑈) ⊆ 𝑈 ∧ (𝑇 ∩ 𝑈) ≠ 𝑈)) | |
25 | 22, 23, 24 | 3bitr4g 314 | 1 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∩ cin 3940 ⊆ wss 3941 ⊊ wpss 3942 ‘cfv 6534 (class class class)co 7402 SubGrpcsubg 19039 LSSumclsm 19546 LModclmod 20698 LSubSpclss 20770 ⋖L clcv 38382 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-lsm 19548 df-mgp 20032 df-ur 20079 df-ring 20132 df-lmod 20700 df-lss 20771 |
This theorem is referenced by: lcvexchlem4 38401 lcvexchlem5 38402 |
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