lcfrlem40 - Mathbox for Norm Megill

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Theorem lcfrlem40 38878

Description: Lemma for lcfr 38881. Eliminate 𝐵 and 𝐼. (Contributed by NM, 11-Mar-2015.)

**Hypotheses**
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 = (0_g‘𝑈)
lcfrlem38.x	⊢ (𝜑 → 𝑋 ≠ 0 )
lcfrlem38.y	⊢ (𝜑 → 𝑌 ≠ 0 )
lcfrlem38.sp	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem38.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
lcfrlem40	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Distinct variable groups: 𝐷,𝑔 𝑔,𝐺 𝑓,𝑔,𝐿 ⊥ ,𝑓,𝑔 + ,𝑓,𝑔 𝑈,𝑓,𝑔 𝑓,𝑋,𝑔 𝑓,𝑌,𝑔 0 ,𝑓,𝑔 𝜑,𝑔 𝑔,𝑁 Allowed substitution hints: 𝜑(𝑓) 𝐶(𝑓,𝑔) 𝐷(𝑓) 𝑄(𝑓,𝑔) 𝐸(𝑓,𝑔) 𝐹(𝑓,𝑔) 𝐺(𝑓) 𝐻(𝑓,𝑔) 𝐾(𝑓,𝑔) 𝑁(𝑓) 𝑊(𝑓,𝑔)

Proof of Theorem lcfrlem40 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcfrlem38.z	. . 3 ⊢ 0 = (0_g‘𝑈)
2		eqid 2798	. . 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 38406	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		lcfrlem38.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
8		eqid 2798	. . . 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 38841	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈))
18		lcfrlem38.x	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
19		eldifsn 4680	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑋 ∈ (Base‘𝑈) ∧ 𝑋 ≠ 0 ))
20	17, 18, 19	sylanbrc 586	. . . 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 38841	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈))
23		lcfrlem38.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 )
24		eldifsn 4680	. . . . 5 ⊢ (𝑌 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑌 ∈ (Base‘𝑈) ∧ 𝑌 ≠ 0 ))
25	22, 23, 24	sylanbrc 586	. . . 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 38859	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ (LSAtoms‘𝑈))
28	1, 2, 6, 27	lsateln0 36291	. 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 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	15	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ∈ 𝑄)
33		lcfrlem38.gs	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
34	33	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ⊆ 𝐶)
35	16	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ∈ 𝐸)
36	21	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ∈ 𝐸)
37	18	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ≠ 0 )
38	23	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ≠ 0 )
39	26	3ad2ant1 1130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
40		eqid 2798	. . . 4 ⊢ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
41		simp2 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
42		simp3 1135	. . . 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 38877	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑋 + 𝑌) ∈ 𝐸)
44	43	rexlimdv3a 3245	. 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 {crab 3110 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 {csn 4525 {cpr 4527 ∪ ciun 4881 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +_gcplusg 16557 0_gc0g 16705 LSubSpclss 19696 LSpanclspn 19736 LSAtomsclsa 36270 LFnlclfn 36353 LKerclk 36381 LDualcld 36419 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-of 7389 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-lfl 36354 df-lkr 36382 df-ldual 36420 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: lcfrlem41 38879

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