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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem8 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 38596. The contradiction of dochexmidlem6 38593 and dochexmidlem7 38594 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 3026 | . 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 38238 | . . . . . 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 19700 | . . . . . . . 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 38482 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆) |
13 | 4, 10, 12 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆) |
14 | dochexmidlem1.p | . . . . . . 7 ⊢ ⊕ = (LSSum‘𝑈) | |
15 | 8, 14 | lsmcl 19847 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘𝑋) ∈ 𝑆) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆) |
16 | 5, 6, 13, 15 | syl3anc 1366 | . . . . 5 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆) |
17 | 7, 8 | lssss 19700 | . . . . 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 483 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → 𝑈 ∈ LMod) |
21 | 16 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆) |
22 | 7, 8 | lss1 19702 | . . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑉 ∈ 𝑆) |
23 | 5, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑆) |
24 | 23 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → 𝑉 ∈ 𝑆) |
25 | df-pss 3952 | . . . . . . . . 9 ⊢ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉 ↔ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) | |
26 | 25 | biimpri 230 | . . . . . . . 8 ⊢ (((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉) |
27 | 26 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉) |
28 | 8, 19, 20, 21, 24, 27 | lpssat 36141 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))) |
29 | 28 | ex 415 | . . . . 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 1128 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | 6 | 3ad2ant1 1128 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑋 ∈ 𝑆) |
33 | simp2 1132 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑝 ∈ 𝐴) | |
34 | dochexmidlem8.z | . . . . . . . . 9 ⊢ 0 = (0g‘𝑈) | |
35 | eqid 2819 | . . . . . . . . 9 ⊢ (𝑋 ⊕ 𝑝) = (𝑋 ⊕ 𝑝) | |
36 | dochexmidlem8.xn | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ { 0 }) | |
37 | 36 | 3ad2ant1 1128 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑋 ≠ { 0 }) |
38 | dochexmidlem8.c | . . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) | |
39 | 38 | 3ad2ant1 1128 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
40 | simp3 1133 | . . . . . . . . 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 38593 | . . . . . . . 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 38594 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 ⊕ 𝑝) ≠ 𝑋) |
43 | 41, 42 | pm2.21ddne 3099 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)) |
44 | 43 | 3adant3l 1175 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)) |
45 | 44 | rexlimdv3a 3284 | . . . . 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 693 | . . 3 ⊢ (𝜑 → ((𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉 → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))) |
48 | 47 | necon1bd 3032 | . 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 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∃wrex 3137 ⊆ wss 3934 ⊊ wpss 3935 {csn 4559 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 0gc0g 16705 LSSumclsm 18751 LModclmod 19626 LSubSpclss 19695 LSpanclspn 19735 LSAtomsclsa 36102 HLchlt 36478 LHypclh 37112 DVecHcdvh 38206 ocHcoch 38475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-riotaBAD 36081 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-undef 7931 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 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 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-dvr 19425 df-drng 19496 df-lmod 19628 df-lss 19696 df-lsp 19736 df-lvec 19867 df-lsatoms 36104 df-lcv 36147 df-oposet 36304 df-ol 36306 df-oml 36307 df-covers 36394 df-ats 36395 df-atl 36426 df-cvlat 36450 df-hlat 36479 df-llines 36626 df-lplanes 36627 df-lvols 36628 df-lines 36629 df-psubsp 36631 df-pmap 36632 df-padd 36924 df-lhyp 37116 df-laut 37117 df-ldil 37232 df-ltrn 37233 df-trl 37287 df-tgrp 37871 df-tendo 37883 df-edring 37885 df-dveca 38131 df-disoa 38157 df-dvech 38207 df-dib 38267 df-dic 38301 df-dih 38357 df-doch 38476 df-djh 38523 |
This theorem is referenced by: dochexmid 38596 |
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