lcfrlem40 - Mathbox for Norm Megill

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Theorem lcfrlem40 41562

Description: Lemma for lcfr 41565. 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 2736	. . 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 41090	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		lcfrlem38.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
8		eqid 2736	. . . 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 41525	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈))
18		lcfrlem38.x	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
19		eldifsn 4784	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑋 ∈ (Base‘𝑈) ∧ 𝑋 ≠ 0 ))
20	17, 18, 19	sylanbrc 583	. . . 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 41525	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈))
23		lcfrlem38.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 )
24		eldifsn 4784	. . . . 5 ⊢ (𝑌 ∈ ((Base‘𝑈) ∖ { 0 }) ↔ (𝑌 ∈ (Base‘𝑈) ∧ 𝑌 ≠ 0 ))
25	22, 23, 24	sylanbrc 583	. . . 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 41543	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ (LSAtoms‘𝑈))
28	1, 2, 6, 27	lsateln0 38974	. 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 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	15	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ∈ 𝑄)
33		lcfrlem38.gs	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
34	33	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝐺 ⊆ 𝐶)
35	16	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ∈ 𝐸)
36	21	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ∈ 𝐸)
37	18	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑋 ≠ 0 )
38	23	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑌 ≠ 0 )
39	26	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
40		eqid 2736	. . . 4 ⊢ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
41		simp2 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
42		simp3 1139	. . . 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 41561	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∧ 𝑖 ≠ 0 ) → (𝑋 + 𝑌) ∈ 𝐸)
44	43	rexlimdv3a 3158	. 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ∃wrex 3069 {crab 3435 ∖ cdif 3947 ∩ cin 3949 ⊆ wss 3950 {csn 4624 {cpr 4626 ∪ ciun 4989 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 +_gcplusg 17293 0_gc0g 17480 LSubSpclss 20921 LSpanclspn 20961 LSAtomsclsa 38953 LFnlclfn 39036 LKerclk 39064 LDualcld 39102 HLchlt 39329 LHypclh 39964 DVecHcdvh 41058 ocHcoch 41327

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-riotaBAD 38932

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-om 7884 df-1st 8010 df-2nd 8011 df-tpos 8247 df-undef 8294 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-er 8741 df-map 8864 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-n0 12523 df-z 12610 df-uz 12875 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-sca 17309 df-vsca 17310 df-0g 17482 df-mre 17625 df-mrc 17626 df-acs 17628 df-proset 18336 df-poset 18355 df-plt 18371 df-lub 18387 df-glb 18388 df-join 18389 df-meet 18390 df-p0 18466 df-p1 18467 df-lat 18473 df-clat 18540 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-submnd 18793 df-grp 18950 df-minusg 18951 df-sbg 18952 df-subg 19137 df-cntz 19331 df-oppg 19360 df-lsm 19650 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-ring 20228 df-oppr 20326 df-dvdsr 20349 df-unit 20350 df-invr 20380 df-dvr 20393 df-nzr 20505 df-rlreg 20686 df-domn 20687 df-drng 20723 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21094 df-lsatoms 38955 df-lshyp 38956 df-lcv 38998 df-lfl 39037 df-lkr 39065 df-ldual 39103 df-oposet 39155 df-ol 39157 df-oml 39158 df-covers 39245 df-ats 39246 df-atl 39277 df-cvlat 39301 df-hlat 39330 df-llines 39478 df-lplanes 39479 df-lvols 39480 df-lines 39481 df-psubsp 39483 df-pmap 39484 df-padd 39776 df-lhyp 39968 df-laut 39969 df-ldil 40084 df-ltrn 40085 df-trl 40139 df-tgrp 40723 df-tendo 40735 df-edring 40737 df-dveca 40983 df-disoa 41009 df-dvech 41059 df-dib 41119 df-dic 41153 df-dih 41209 df-doch 41328 df-djh 41375

This theorem is referenced by: lcfrlem41 41563

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