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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem8 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 38764. The contradiction of dochexmidlem6 38761 and dochexmidlem7 38762 shows that there can be no atom 𝑝 that is not in 𝑋 + ( ⊥ ‘𝑋), which is therefore the whole atom space. (Contributed by NM, 15-Jan-2015.) |
Ref | Expression |
---|---|
dochexmidlem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochexmidlem1.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochexmidlem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochexmidlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
dochexmidlem1.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
dochexmidlem1.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dochexmidlem1.p | ⊢ ⊕ = (LSSum‘𝑈) |
dochexmidlem1.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
dochexmidlem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochexmidlem1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
dochexmidlem8.z | ⊢ 0 = (0g‘𝑈) |
dochexmidlem8.xn | ⊢ (𝜑 → 𝑋 ≠ { 0 }) |
dochexmidlem8.c | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
Ref | Expression |
---|---|
dochexmidlem8 | ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nonconne 2999 | . 2 ⊢ ¬ (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋) | |
2 | dochexmidlem1.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | dochexmidlem1.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | dochexmidlem1.k | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
5 | 2, 3, 4 | dvhlmod 38406 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
6 | dochexmidlem1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
7 | dochexmidlem1.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈) | |
8 | dochexmidlem1.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑈) | |
9 | 7, 8 | lssss 19701 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉) |
10 | 6, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
11 | dochexmidlem1.o | . . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
12 | 2, 3, 7, 8, 11 | dochlss 38650 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆) |
13 | 4, 10, 12 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆) |
14 | dochexmidlem1.p | . . . . . . 7 ⊢ ⊕ = (LSSum‘𝑈) | |
15 | 8, 14 | lsmcl 19848 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘𝑋) ∈ 𝑆) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆) |
16 | 5, 6, 13, 15 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆) |
17 | 7, 8 | lssss 19701 | . . . . 5 ⊢ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉) |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉) |
19 | dochexmidlem1.a | . . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
20 | 5 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → 𝑈 ∈ LMod) |
21 | 16 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆) |
22 | 7, 8 | lss1 19703 | . . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑉 ∈ 𝑆) |
23 | 5, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑆) |
24 | 23 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → 𝑉 ∈ 𝑆) |
25 | df-pss 3900 | . . . . . . . . 9 ⊢ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉 ↔ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) | |
26 | 25 | biimpri 231 | . . . . . . . 8 ⊢ (((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉) |
27 | 26 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉) |
28 | 8, 19, 20, 21, 24, 27 | lpssat 36309 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))) |
29 | 28 | ex 416 | . . . . 5 ⊢ (𝜑 → (((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))))) |
30 | dochexmidlem1.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈) | |
31 | 4 | 3ad2ant1 1130 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | 6 | 3ad2ant1 1130 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑋 ∈ 𝑆) |
33 | simp2 1134 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑝 ∈ 𝐴) | |
34 | dochexmidlem8.z | . . . . . . . . 9 ⊢ 0 = (0g‘𝑈) | |
35 | eqid 2798 | . . . . . . . . 9 ⊢ (𝑋 ⊕ 𝑝) = (𝑋 ⊕ 𝑝) | |
36 | dochexmidlem8.xn | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ { 0 }) | |
37 | 36 | 3ad2ant1 1130 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑋 ≠ { 0 }) |
38 | dochexmidlem8.c | . . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) | |
39 | 38 | 3ad2ant1 1130 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
40 | simp3 1135 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) | |
41 | 2, 11, 3, 7, 8, 30, 14, 19, 31, 32, 33, 34, 35, 37, 39, 40 | dochexmidlem6 38761 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 ⊕ 𝑝) = 𝑋) |
42 | 2, 11, 3, 7, 8, 30, 14, 19, 31, 32, 33, 34, 35, 37, 39, 40 | dochexmidlem7 38762 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 ⊕ 𝑝) ≠ 𝑋) |
43 | 41, 42 | pm2.21ddne 3071 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)) |
44 | 43 | 3adant3l 1177 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)) |
45 | 44 | rexlimdv3a 3245 | . . . . 5 ⊢ (𝜑 → (∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))) |
46 | 29, 45 | syld 47 | . . . 4 ⊢ (𝜑 → (((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))) |
47 | 18, 46 | mpand 694 | . . 3 ⊢ (𝜑 → ((𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉 → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))) |
48 | 47 | necon1bd 3005 | . 2 ⊢ (𝜑 → (¬ (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋) → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉)) |
49 | 1, 48 | mpi 20 | 1 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ⊆ wss 3881 ⊊ wpss 3882 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 0gc0g 16705 LSSumclsm 18751 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 LSAtomsclsa 36270 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 ocHcoch 38643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lcv 36315 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tgrp 38039 df-tendo 38051 df-edring 38053 df-dveca 38299 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 df-djh 38691 |
This theorem is referenced by: dochexmid 38764 |
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