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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem21 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 38881. (Contributed by NM, 11-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem17.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfrlem17.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfrlem17.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfrlem17.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfrlem17.p | ⊢ + = (+g‘𝑈) |
lcfrlem17.z | ⊢ 0 = (0g‘𝑈) |
lcfrlem17.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfrlem17.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
lcfrlem17.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem17.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lcfrlem17.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lcfrlem17.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lcfrlem21 | ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem17.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcfrlem17.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lcfrlem17.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lcfrlem17.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lcfrlem17.p | . . 3 ⊢ + = (+g‘𝑈) | |
6 | lcfrlem17.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
7 | lcfrlem17.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
8 | lcfrlem17.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
9 | lcfrlem17.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 9 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
11 | lcfrlem17.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
12 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
13 | lcfrlem17.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
14 | 13 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
15 | lcfrlem17.ne | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
16 | 15 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
17 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17 | lcfrlem20 38858 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴) |
19 | 1, 3, 9 | dvhlmod 38406 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
20 | 11 | eldifad 3893 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
21 | 13 | eldifad 3893 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
22 | 4, 5 | lmodcom 19673 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
23 | 19, 20, 21, 22 | syl3anc 1368 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
24 | 23 | sneqd 4537 | . . . . . . 7 ⊢ (𝜑 → {(𝑋 + 𝑌)} = {(𝑌 + 𝑋)}) |
25 | 24 | fveq2d 6649 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = ( ⊥ ‘{(𝑌 + 𝑋)})) |
26 | 25 | eleq2d 2875 | . . . . 5 ⊢ (𝜑 → (𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ↔ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}))) |
27 | 26 | biimprd 251 | . . . 4 ⊢ (𝜑 → (𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}) → 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))) |
28 | 27 | con3dimp 412 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) |
29 | prcom 4628 | . . . . . . . 8 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
30 | 29 | fveq2i 6648 | . . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋}) |
31 | 30 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋})) |
32 | 31, 25 | ineq12d 4140 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)}))) |
33 | 32 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)}))) |
34 | 9 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
35 | 13 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
36 | 11 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
37 | 15 | necomd 3042 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋})) |
38 | 37 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋})) |
39 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) | |
40 | 1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 38, 39 | lcfrlem20 38858 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)})) ∈ 𝐴) |
41 | 33, 40 | eqeltrd 2890 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴) |
42 | 28, 41 | syldan 594 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴) |
43 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15 | lcfrlem19 38857 | . 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))) |
44 | 18, 42, 43 | mpjaodan 956 | 1 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 ∩ cin 3880 {csn 4525 {cpr 4527 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 LModclmod 19627 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-lshyp 36273 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: lcfrlem22 38860 lcfrlem40 38878 |
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