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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2c | Structured version Visualization version GIF version |
Description: Lemma for lclkr 40898. (Contributed by NM, 16-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrlem2a.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrlem2a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrlem2a.v | ⊢ 𝑉 = (Base‘𝑈) |
lclkrlem2a.z | ⊢ 0 = (0g‘𝑈) |
lclkrlem2a.p | ⊢ ⊕ = (LSSum‘𝑈) |
lclkrlem2a.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lclkrlem2a.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
lclkrlem2a.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrlem2a.b | ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2a.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2a.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2a.e | ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌})) |
lclkrlem2b.da | ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) |
lclkrlem2c.j | ⊢ 𝐽 = (LSHyp‘𝑈) |
Ref | Expression |
---|---|
lclkrlem2c | ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2a.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2724 | . . . 4 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
3 | lclkrlem2a.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lclkrlem2a.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lclkrlem2a.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
6 | eqid 2724 | . . . 4 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊) | |
7 | lclkrlem2a.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | lclkrlem2a.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝑈) | |
9 | lclkrlem2a.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
10 | lclkrlem2a.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
11 | 10 | eldifad 3953 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
12 | lclkrlem2a.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈) | |
13 | 1, 3, 4, 12, 2 | dihlsprn 40696 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
14 | 7, 11, 13 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
15 | lclkrlem2a.z | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
16 | 1, 3, 7 | dvhlmod 40475 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
17 | lclkrlem2a.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
18 | 4, 12, 15, 9, 16, 17 | lsatlspsn 38357 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴) |
19 | 1, 2, 3, 8, 9, 7, 14, 18 | dihsmatrn 40801 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
20 | lclkrlem2a.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) | |
21 | 20 | eldifad 3953 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
22 | 21 | snssd 4805 | . . . . 5 ⊢ (𝜑 → {𝐵} ⊆ 𝑉) |
23 | 1, 2, 3, 4, 5 | dochcl 40718 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐵} ⊆ 𝑉) → ( ⊥ ‘{𝐵}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
24 | 7, 22, 23 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
25 | 1, 2, 3, 4, 5, 6, 7, 19, 24 | dochdmm1 40775 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵})))) |
26 | 17 | eldifad 3953 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
27 | 4, 12, 8, 16, 11, 26 | lsmpr 20929 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) |
28 | df-pr 4624 | . . . . . . . 8 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
29 | 28 | fveq2i 6885 | . . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
30 | 27, 29 | eqtr3di 2779 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌}))) |
31 | 30 | fveq2d 6886 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌})))) |
32 | 11 | snssd 4805 | . . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
33 | 26 | snssd 4805 | . . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
34 | 32, 33 | unssd 4179 | . . . . . 6 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ 𝑉) |
35 | 1, 3, 5, 4, 12, 7, 34 | dochocsp 40744 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘({𝑋} ∪ {𝑌}))) = ( ⊥ ‘({𝑋} ∪ {𝑌}))) |
36 | 1, 3, 4, 5 | dochdmj1 40755 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))) |
37 | 7, 32, 33, 36 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘({𝑋} ∪ {𝑌})) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))) |
38 | 31, 35, 37 | 3eqtrd 2768 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))) |
39 | 1, 3, 5, 4, 12, 7, 21 | dochocsn 40746 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝐵})) = (𝑁‘{𝐵})) |
40 | 38, 39 | oveq12d 7420 | . . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{𝐵}))) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵}))) |
41 | 1, 2, 3, 4, 5 | dochcl 40718 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
42 | 7, 32, 41 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
43 | 1, 2, 3, 4, 5 | dochcl 40718 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
44 | 7, 33, 43 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
45 | 1, 2 | dihmeetcl 40710 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊) ∧ ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))) → (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
46 | 7, 42, 44, 45 | syl12anc 834 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
47 | 1, 3, 4, 8, 12, 2, 6, 7, 46, 21 | dihjat1 40794 | . . 3 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{𝐵})) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵}))) |
48 | 25, 40, 47 | 3eqtrrd 2769 | . 2 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) = ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})))) |
49 | lclkrlem2c.j | . . 3 ⊢ 𝐽 = (LSHyp‘𝑈) | |
50 | lclkrlem2a.e | . . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌})) | |
51 | lclkrlem2b.da | . . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) | |
52 | 1, 5, 3, 4, 15, 8, 12, 9, 7, 20, 10, 17, 50, 51 | lclkrlem2b 40873 | . . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴) |
53 | 1, 3, 5, 9, 49, 7, 52 | dochsatshp 40816 | . 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵}))) ∈ 𝐽) |
54 | 48, 53 | eqeltrd 2825 | 1 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3938 ∪ cun 3939 ∩ cin 3940 ⊆ wss 3941 {csn 4621 {cpr 4623 ran crn 5668 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 0gc0g 17386 LSSumclsm 19546 LSpanclspn 20810 LSAtomsclsa 38338 LSHypclsh 38339 HLchlt 38714 LHypclh 39349 DVecHcdvh 40443 DIsoHcdih 40593 ocHcoch 40712 joinHcdjh 40759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38317 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-0g 17388 df-mre 17531 df-mrc 17532 df-acs 17534 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-cntz 19225 df-oppg 19254 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-invr 20282 df-dvr 20295 df-drng 20581 df-lmod 20700 df-lss 20771 df-lsp 20811 df-lvec 20943 df-lsatoms 38340 df-lshyp 38341 df-lcv 38383 df-oposet 38540 df-ol 38542 df-oml 38543 df-covers 38630 df-ats 38631 df-atl 38662 df-cvlat 38686 df-hlat 38715 df-llines 38863 df-lplanes 38864 df-lvols 38865 df-lines 38866 df-psubsp 38868 df-pmap 38869 df-padd 39161 df-lhyp 39353 df-laut 39354 df-ldil 39469 df-ltrn 39470 df-trl 39524 df-tgrp 40108 df-tendo 40120 df-edring 40122 df-dveca 40368 df-disoa 40394 df-dvech 40444 df-dib 40504 df-dic 40538 df-dih 40594 df-doch 40713 df-djh 40760 |
This theorem is referenced by: lclkrlem2g 40878 |
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