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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem40 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 38736. Eliminate 𝐵 and 𝐼. (Contributed by NM, 11-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem38.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfrlem38.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfrlem38.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfrlem38.p | ⊢ + = (+g‘𝑈) |
lcfrlem38.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfrlem38.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfrlem38.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcfrlem38.q | ⊢ 𝑄 = (LSubSp‘𝐷) |
lcfrlem38.c | ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcfrlem38.e | ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) |
lcfrlem38.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem38.g | ⊢ (𝜑 → 𝐺 ∈ 𝑄) |
lcfrlem38.gs | ⊢ (𝜑 → 𝐺 ⊆ 𝐶) |
lcfrlem38.xe | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
lcfrlem38.ye | ⊢ (𝜑 → 𝑌 ∈ 𝐸) |
lcfrlem38.z | ⊢ 0 = (0g‘𝑈) |
lcfrlem38.x | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
lcfrlem38.y | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
lcfrlem38.sp | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfrlem38.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lcfrlem40 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem38.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
2 | eqid 2821 | . . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
3 | lcfrlem38.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | lcfrlem38.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | lcfrlem38.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 3, 4, 5 | dvhlmod 38261 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
7 | lcfrlem38.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
8 | eqid 2821 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
9 | lcfrlem38.p | . . . 4 ⊢ + = (+g‘𝑈) | |
10 | lcfrlem38.sp | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
11 | lcfrlem38.l | . . . . . 6 ⊢ 𝐿 = (LKer‘𝑈) | |
12 | lcfrlem38.d | . . . . . 6 ⊢ 𝐷 = (LDual‘𝑈) | |
13 | lcfrlem38.q | . . . . . 6 ⊢ 𝑄 = (LSubSp‘𝐷) | |
14 | lcfrlem38.e | . . . . . 6 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) | |
15 | lcfrlem38.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑄) | |
16 | lcfrlem38.xe | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
17 | 3, 7, 4, 8, 11, 12, 13, 14, 5, 15, 16 | lcfrlem4 38696 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈)) |
18 | lcfrlem38.x | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
19 | eldifsn 4719 | . . . . 5 ⊢ (𝑋 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑋 ∈ (Base‘𝑈) ∧ 𝑋 ≠ 0 )) | |
20 | 17, 18, 19 | sylanbrc 585 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑈) ∖ { 0 })) |
21 | lcfrlem38.ye | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐸) | |
22 | 3, 7, 4, 8, 11, 12, 13, 14, 5, 15, 21 | lcfrlem4 38696 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈)) |
23 | lcfrlem38.y | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
24 | eldifsn 4719 | . . . . 5 ⊢ (𝑌 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑌 ∈ (Base‘𝑈) ∧ 𝑌 ≠ 0 )) | |
25 | 22, 23, 24 | sylanbrc 585 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝑈) ∖ { 0 })) |
26 | lcfrlem38.ne | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
27 | 3, 7, 4, 8, 9, 1, 10, 2, 5, 20, 25, 26 | lcfrlem21 38714 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ (LSAtoms‘𝑈)) |
28 | 1, 2, 6, 27 | lsateln0 36146 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))𝑖 ≠ 0 ) |
29 | lcfrlem38.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
30 | lcfrlem38.c | . . . 4 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
31 | 5 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | 15 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ∈ 𝑄) |
33 | lcfrlem38.gs | . . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶) | |
34 | 33 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ⊆ 𝐶) |
35 | 16 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ∈ 𝐸) |
36 | 21 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ∈ 𝐸) |
37 | 18 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ≠ 0 ) |
38 | 23 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ≠ 0 ) |
39 | 26 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
40 | eqid 2821 | . . . 4 ⊢ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
41 | simp2 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) | |
42 | simp3 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑖 ≠ 0 ) | |
43 | 3, 7, 4, 9, 29, 11, 12, 13, 30, 14, 31, 32, 34, 35, 36, 1, 37, 38, 10, 39, 40, 41, 42 | lcfrlem39 38732 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑋 + 𝑌) ∈ 𝐸) |
44 | 43 | rexlimdv3a 3286 | . 2 ⊢ (𝜑 → (∃𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))𝑖 ≠ 0 → (𝑋 + 𝑌) ∈ 𝐸)) |
45 | 28, 44 | mpd 15 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {crab 3142 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 {csn 4567 {cpr 4569 ∪ ciun 4919 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 LSubSpclss 19703 LSpanclspn 19743 LSAtomsclsa 36125 LFnlclfn 36208 LKerclk 36236 LDualcld 36274 HLchlt 36501 LHypclh 37135 DVecHcdvh 38229 ocHcoch 38498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-mre 16857 df-mrc 16858 df-acs 16860 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-oppg 18474 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lsatoms 36127 df-lshyp 36128 df-lcv 36170 df-lfl 36209 df-lkr 36237 df-ldual 36275 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tgrp 37894 df-tendo 37906 df-edring 37908 df-dveca 38154 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 df-doch 38499 df-djh 38546 |
This theorem is referenced by: lcfrlem41 38734 |
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