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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvexchlem1 | Structured version Visualization version GIF version |
Description: Lemma for lcvexch 36676. (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 19849 | . . . . . . 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 3878 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
7 | lcvexch.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 4, 7 | sseldd 3878 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
9 | lcvexch.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
10 | 9 | lsmub1 18900 | . . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) |
11 | 6, 8, 10 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) |
12 | inss2 4120 | . . . . 5 ⊢ (𝑇 ∩ 𝑈) ⊆ 𝑈 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑈) |
14 | 11, 13 | 2thd 268 | . . 3 ⊢ (𝜑 → (𝑇 ⊆ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊆ 𝑈)) |
15 | 9 | lsmss2b 18912 | . . . . . . 7 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) |
16 | 6, 8, 15 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) |
17 | eqcom 2745 | . . . . . 6 ⊢ ((𝑇 ⊕ 𝑈) = 𝑇 ↔ 𝑇 = (𝑇 ⊕ 𝑈)) | |
18 | 16, 17 | bitrdi 290 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊆ 𝑇 ↔ 𝑇 = (𝑇 ⊕ 𝑈))) |
19 | sseqin2 4106 | . . . . 5 ⊢ (𝑈 ⊆ 𝑇 ↔ (𝑇 ∩ 𝑈) = 𝑈) | |
20 | 18, 19 | bitr3di 289 | . . . 4 ⊢ (𝜑 → (𝑇 = (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) = 𝑈)) |
21 | 20 | necon3bid 2978 | . . 3 ⊢ (𝜑 → (𝑇 ≠ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ≠ 𝑈)) |
22 | 14, 21 | anbi12d 634 | . 2 ⊢ (𝜑 → ((𝑇 ⊆ (𝑇 ⊕ 𝑈) ∧ 𝑇 ≠ (𝑇 ⊕ 𝑈)) ↔ ((𝑇 ∩ 𝑈) ⊆ 𝑈 ∧ (𝑇 ∩ 𝑈) ≠ 𝑈))) |
23 | df-pss 3862 | . 2 ⊢ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ⊆ (𝑇 ⊕ 𝑈) ∧ 𝑇 ≠ (𝑇 ⊕ 𝑈))) | |
24 | df-pss 3862 | . 2 ⊢ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ↔ ((𝑇 ∩ 𝑈) ⊆ 𝑈 ∧ (𝑇 ∩ 𝑈) ≠ 𝑈)) | |
25 | 22, 23, 24 | 3bitr4g 317 | 1 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∩ cin 3842 ⊆ wss 3843 ⊊ wpss 3844 ‘cfv 6339 (class class class)co 7170 SubGrpcsubg 18391 LSSumclsm 18877 LModclmod 19753 LSubSpclss 19822 ⋖L clcv 36655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-lsm 18879 df-mgp 19359 df-ur 19371 df-ring 19418 df-lmod 19755 df-lss 19823 |
This theorem is referenced by: lcvexchlem4 36674 lcvexchlem5 36675 |
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