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Theorem lcvexchlem5 33343
Description: Lemma for lcvexch 33344. (Contributed by NM, 10-Jan-2015.)
Hypotheses
Ref Expression
lcvexch.s 𝑆 = (LSubSp‘𝑊)
lcvexch.p = (LSSum‘𝑊)
lcvexch.c 𝐶 = ( ⋖L𝑊)
lcvexch.w (𝜑𝑊 ∈ LMod)
lcvexch.t (𝜑𝑇𝑆)
lcvexch.u (𝜑𝑈𝑆)
lcvexch.g (𝜑 → (𝑇𝑈)𝐶𝑈)
Assertion
Ref Expression
lcvexchlem5 (𝜑𝑇𝐶(𝑇 𝑈))

Proof of Theorem lcvexchlem5
Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lcvexch.s . . . 4 𝑆 = (LSubSp‘𝑊)
2 lcvexch.c . . . 4 𝐶 = ( ⋖L𝑊)
3 lcvexch.w . . . 4 (𝜑𝑊 ∈ LMod)
4 lcvexch.t . . . . 5 (𝜑𝑇𝑆)
5 lcvexch.u . . . . 5 (𝜑𝑈𝑆)
61lssincl 18786 . . . . 5 ((𝑊 ∈ LMod ∧ 𝑇𝑆𝑈𝑆) → (𝑇𝑈) ∈ 𝑆)
73, 4, 5, 6syl3anc 1318 . . . 4 (𝜑 → (𝑇𝑈) ∈ 𝑆)
8 lcvexch.g . . . 4 (𝜑 → (𝑇𝑈)𝐶𝑈)
91, 2, 3, 7, 5, 8lcvpss 33329 . . 3 (𝜑 → (𝑇𝑈) ⊊ 𝑈)
10 lcvexch.p . . . 4 = (LSSum‘𝑊)
111, 10, 2, 3, 4, 5lcvexchlem1 33339 . . 3 (𝜑 → (𝑇 ⊊ (𝑇 𝑈) ↔ (𝑇𝑈) ⊊ 𝑈))
129, 11mpbird 246 . 2 (𝜑𝑇 ⊊ (𝑇 𝑈))
13 simp3l 1082 . . . . . . . 8 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → 𝑇𝑠)
14 ssrin 3800 . . . . . . . 8 (𝑇𝑠 → (𝑇𝑈) ⊆ (𝑠𝑈))
1513, 14syl 17 . . . . . . 7 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → (𝑇𝑈) ⊆ (𝑠𝑈))
16 inss2 3796 . . . . . . 7 (𝑠𝑈) ⊆ 𝑈
1715, 16jctir 559 . . . . . 6 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → ((𝑇𝑈) ⊆ (𝑠𝑈) ∧ (𝑠𝑈) ⊆ 𝑈))
1883ad2ant1 1075 . . . . . . 7 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → (𝑇𝑈)𝐶𝑈)
191, 2, 3, 7, 5lcvbr3 33328 . . . . . . . . . 10 (𝜑 → ((𝑇𝑈)𝐶𝑈 ↔ ((𝑇𝑈) ⊊ 𝑈 ∧ ∀𝑟𝑆 (((𝑇𝑈) ⊆ 𝑟𝑟𝑈) → (𝑟 = (𝑇𝑈) ∨ 𝑟 = 𝑈)))))
2019adantr 480 . . . . . . . . 9 ((𝜑𝑠𝑆) → ((𝑇𝑈)𝐶𝑈 ↔ ((𝑇𝑈) ⊊ 𝑈 ∧ ∀𝑟𝑆 (((𝑇𝑈) ⊆ 𝑟𝑟𝑈) → (𝑟 = (𝑇𝑈) ∨ 𝑟 = 𝑈)))))
213adantr 480 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → 𝑊 ∈ LMod)
22 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → 𝑠𝑆)
235adantr 480 . . . . . . . . . . . 12 ((𝜑𝑠𝑆) → 𝑈𝑆)
241lssincl 18786 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ 𝑠𝑆𝑈𝑆) → (𝑠𝑈) ∈ 𝑆)
2521, 22, 23, 24syl3anc 1318 . . . . . . . . . . 11 ((𝜑𝑠𝑆) → (𝑠𝑈) ∈ 𝑆)
26 sseq2 3590 . . . . . . . . . . . . . 14 (𝑟 = (𝑠𝑈) → ((𝑇𝑈) ⊆ 𝑟 ↔ (𝑇𝑈) ⊆ (𝑠𝑈)))
27 sseq1 3589 . . . . . . . . . . . . . 14 (𝑟 = (𝑠𝑈) → (𝑟𝑈 ↔ (𝑠𝑈) ⊆ 𝑈))
2826, 27anbi12d 743 . . . . . . . . . . . . 13 (𝑟 = (𝑠𝑈) → (((𝑇𝑈) ⊆ 𝑟𝑟𝑈) ↔ ((𝑇𝑈) ⊆ (𝑠𝑈) ∧ (𝑠𝑈) ⊆ 𝑈)))
29 eqeq1 2614 . . . . . . . . . . . . . 14 (𝑟 = (𝑠𝑈) → (𝑟 = (𝑇𝑈) ↔ (𝑠𝑈) = (𝑇𝑈)))
30 eqeq1 2614 . . . . . . . . . . . . . 14 (𝑟 = (𝑠𝑈) → (𝑟 = 𝑈 ↔ (𝑠𝑈) = 𝑈))
3129, 30orbi12d 742 . . . . . . . . . . . . 13 (𝑟 = (𝑠𝑈) → ((𝑟 = (𝑇𝑈) ∨ 𝑟 = 𝑈) ↔ ((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈)))
3228, 31imbi12d 333 . . . . . . . . . . . 12 (𝑟 = (𝑠𝑈) → ((((𝑇𝑈) ⊆ 𝑟𝑟𝑈) → (𝑟 = (𝑇𝑈) ∨ 𝑟 = 𝑈)) ↔ (((𝑇𝑈) ⊆ (𝑠𝑈) ∧ (𝑠𝑈) ⊆ 𝑈) → ((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈))))
3332rspcv 3278 . . . . . . . . . . 11 ((𝑠𝑈) ∈ 𝑆 → (∀𝑟𝑆 (((𝑇𝑈) ⊆ 𝑟𝑟𝑈) → (𝑟 = (𝑇𝑈) ∨ 𝑟 = 𝑈)) → (((𝑇𝑈) ⊆ (𝑠𝑈) ∧ (𝑠𝑈) ⊆ 𝑈) → ((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈))))
3425, 33syl 17 . . . . . . . . . 10 ((𝜑𝑠𝑆) → (∀𝑟𝑆 (((𝑇𝑈) ⊆ 𝑟𝑟𝑈) → (𝑟 = (𝑇𝑈) ∨ 𝑟 = 𝑈)) → (((𝑇𝑈) ⊆ (𝑠𝑈) ∧ (𝑠𝑈) ⊆ 𝑈) → ((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈))))
3534adantld 482 . . . . . . . . 9 ((𝜑𝑠𝑆) → (((𝑇𝑈) ⊊ 𝑈 ∧ ∀𝑟𝑆 (((𝑇𝑈) ⊆ 𝑟𝑟𝑈) → (𝑟 = (𝑇𝑈) ∨ 𝑟 = 𝑈))) → (((𝑇𝑈) ⊆ (𝑠𝑈) ∧ (𝑠𝑈) ⊆ 𝑈) → ((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈))))
3620, 35sylbid 229 . . . . . . . 8 ((𝜑𝑠𝑆) → ((𝑇𝑈)𝐶𝑈 → (((𝑇𝑈) ⊆ (𝑠𝑈) ∧ (𝑠𝑈) ⊆ 𝑈) → ((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈))))
37363adant3 1074 . . . . . . 7 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → ((𝑇𝑈)𝐶𝑈 → (((𝑇𝑈) ⊆ (𝑠𝑈) ∧ (𝑠𝑈) ⊆ 𝑈) → ((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈))))
3818, 37mpd 15 . . . . . 6 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → (((𝑇𝑈) ⊆ (𝑠𝑈) ∧ (𝑠𝑈) ⊆ 𝑈) → ((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈)))
3917, 38mpd 15 . . . . 5 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → ((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈))
40 oveq1 6556 . . . . . . 7 ((𝑠𝑈) = (𝑇𝑈) → ((𝑠𝑈) 𝑇) = ((𝑇𝑈) 𝑇))
4133ad2ant1 1075 . . . . . . . . 9 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → 𝑊 ∈ LMod)
4243ad2ant1 1075 . . . . . . . . 9 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → 𝑇𝑆)
4353ad2ant1 1075 . . . . . . . . 9 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → 𝑈𝑆)
44 simp2 1055 . . . . . . . . 9 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → 𝑠𝑆)
45 simp3r 1083 . . . . . . . . 9 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → 𝑠 ⊆ (𝑇 𝑈))
461, 10, 2, 41, 42, 43, 44, 13, 45lcvexchlem3 33341 . . . . . . . 8 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → ((𝑠𝑈) 𝑇) = 𝑠)
471lsssssubg 18779 . . . . . . . . . . . 12 (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
483, 47syl 17 . . . . . . . . . . 11 (𝜑𝑆 ⊆ (SubGrp‘𝑊))
4948, 7sseldd 3569 . . . . . . . . . 10 (𝜑 → (𝑇𝑈) ∈ (SubGrp‘𝑊))
5048, 4sseldd 3569 . . . . . . . . . 10 (𝜑𝑇 ∈ (SubGrp‘𝑊))
51 inss1 3795 . . . . . . . . . . 11 (𝑇𝑈) ⊆ 𝑇
5251a1i 11 . . . . . . . . . 10 (𝜑 → (𝑇𝑈) ⊆ 𝑇)
5310lsmss1 17902 . . . . . . . . . 10 (((𝑇𝑈) ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ (𝑇𝑈) ⊆ 𝑇) → ((𝑇𝑈) 𝑇) = 𝑇)
5449, 50, 52, 53syl3anc 1318 . . . . . . . . 9 (𝜑 → ((𝑇𝑈) 𝑇) = 𝑇)
55543ad2ant1 1075 . . . . . . . 8 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → ((𝑇𝑈) 𝑇) = 𝑇)
5646, 55eqeq12d 2625 . . . . . . 7 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → (((𝑠𝑈) 𝑇) = ((𝑇𝑈) 𝑇) ↔ 𝑠 = 𝑇))
5740, 56syl5ib 233 . . . . . 6 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → ((𝑠𝑈) = (𝑇𝑈) → 𝑠 = 𝑇))
58 oveq1 6556 . . . . . . 7 ((𝑠𝑈) = 𝑈 → ((𝑠𝑈) 𝑇) = (𝑈 𝑇))
59 lmodabl 18733 . . . . . . . . . . 11 (𝑊 ∈ LMod → 𝑊 ∈ Abel)
603, 59syl 17 . . . . . . . . . 10 (𝜑𝑊 ∈ Abel)
6148, 5sseldd 3569 . . . . . . . . . 10 (𝜑𝑈 ∈ (SubGrp‘𝑊))
6210lsmcom 18084 . . . . . . . . . 10 ((𝑊 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊)) → (𝑈 𝑇) = (𝑇 𝑈))
6360, 61, 50, 62syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝑈 𝑇) = (𝑇 𝑈))
64633ad2ant1 1075 . . . . . . . 8 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → (𝑈 𝑇) = (𝑇 𝑈))
6546, 64eqeq12d 2625 . . . . . . 7 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → (((𝑠𝑈) 𝑇) = (𝑈 𝑇) ↔ 𝑠 = (𝑇 𝑈)))
6658, 65syl5ib 233 . . . . . 6 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → ((𝑠𝑈) = 𝑈𝑠 = (𝑇 𝑈)))
6757, 66orim12d 879 . . . . 5 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → (((𝑠𝑈) = (𝑇𝑈) ∨ (𝑠𝑈) = 𝑈) → (𝑠 = 𝑇𝑠 = (𝑇 𝑈))))
6839, 67mpd 15 . . . 4 ((𝜑𝑠𝑆 ∧ (𝑇𝑠𝑠 ⊆ (𝑇 𝑈))) → (𝑠 = 𝑇𝑠 = (𝑇 𝑈)))
69683exp 1256 . . 3 (𝜑 → (𝑠𝑆 → ((𝑇𝑠𝑠 ⊆ (𝑇 𝑈)) → (𝑠 = 𝑇𝑠 = (𝑇 𝑈)))))
7069ralrimiv 2948 . 2 (𝜑 → ∀𝑠𝑆 ((𝑇𝑠𝑠 ⊆ (𝑇 𝑈)) → (𝑠 = 𝑇𝑠 = (𝑇 𝑈))))
711, 10lsmcl 18904 . . . 4 ((𝑊 ∈ LMod ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) ∈ 𝑆)
723, 4, 5, 71syl3anc 1318 . . 3 (𝜑 → (𝑇 𝑈) ∈ 𝑆)
731, 2, 3, 4, 72lcvbr3 33328 . 2 (𝜑 → (𝑇𝐶(𝑇 𝑈) ↔ (𝑇 ⊊ (𝑇 𝑈) ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠 ⊆ (𝑇 𝑈)) → (𝑠 = 𝑇𝑠 = (𝑇 𝑈))))))
7412, 70, 73mpbir2and 959 1 (𝜑𝑇𝐶(𝑇 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  cin 3539  wss 3540  wpss 3541   class class class wbr 4583  cfv 5804  (class class class)co 6549  SubGrpcsubg 17411  LSSumclsm 17872  Abelcabl 18017  LModclmod 18686  LSubSpclss 18753  L clcv 33323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-cntz 17573  df-oppg 17599  df-lsm 17874  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lss 18754  df-lcv 33324
This theorem is referenced by:  lcvexch  33344
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