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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvexchlem1 | Structured version Visualization version GIF version |
Description: Lemma for lcvexch 35114. (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 19317 | . . . . . . 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 3828 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
7 | lcvexch.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 4, 7 | sseldd 3828 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
9 | lcvexch.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
10 | 9 | lsmub1 18422 | . . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) |
11 | 6, 8, 10 | syl2anc 581 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑇 ⊕ 𝑈)) |
12 | inss2 4058 | . . . . 5 ⊢ (𝑇 ∩ 𝑈) ⊆ 𝑈 | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑈) |
14 | 11, 13 | 2thd 257 | . . 3 ⊢ (𝜑 → (𝑇 ⊆ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊆ 𝑈)) |
15 | sseqin2 4044 | . . . . 5 ⊢ (𝑈 ⊆ 𝑇 ↔ (𝑇 ∩ 𝑈) = 𝑈) | |
16 | 9 | lsmss2b 18433 | . . . . . . 7 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) |
17 | 6, 8, 16 | syl2anc 581 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇)) |
18 | eqcom 2832 | . . . . . 6 ⊢ ((𝑇 ⊕ 𝑈) = 𝑇 ↔ 𝑇 = (𝑇 ⊕ 𝑈)) | |
19 | 17, 18 | syl6bb 279 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊆ 𝑇 ↔ 𝑇 = (𝑇 ⊕ 𝑈))) |
20 | 15, 19 | syl5rbbr 278 | . . . 4 ⊢ (𝜑 → (𝑇 = (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) = 𝑈)) |
21 | 20 | necon3bid 3043 | . . 3 ⊢ (𝜑 → (𝑇 ≠ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ≠ 𝑈)) |
22 | 14, 21 | anbi12d 626 | . 2 ⊢ (𝜑 → ((𝑇 ⊆ (𝑇 ⊕ 𝑈) ∧ 𝑇 ≠ (𝑇 ⊕ 𝑈)) ↔ ((𝑇 ∩ 𝑈) ⊆ 𝑈 ∧ (𝑇 ∩ 𝑈) ≠ 𝑈))) |
23 | df-pss 3814 | . 2 ⊢ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ⊆ (𝑇 ⊕ 𝑈) ∧ 𝑇 ≠ (𝑇 ⊕ 𝑈))) | |
24 | df-pss 3814 | . 2 ⊢ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ↔ ((𝑇 ∩ 𝑈) ⊆ 𝑈 ∧ (𝑇 ∩ 𝑈) ≠ 𝑈)) | |
25 | 22, 23, 24 | 3bitr4g 306 | 1 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∩ cin 3797 ⊆ wss 3798 ⊊ wpss 3799 ‘cfv 6123 (class class class)co 6905 SubGrpcsubg 17939 LSSumclsm 18400 LModclmod 19219 LSubSpclss 19288 ⋖L clcv 35093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-lsm 18402 df-mgp 18844 df-ur 18856 df-ring 18903 df-lmod 19221 df-lss 19289 |
This theorem is referenced by: lcvexchlem4 35112 lcvexchlem5 35113 |
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