lcfrlem41 - Mathbox for Norm Megill

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Theorem lcfrlem41 39334

Description: Lemma for lcfr 39336. Eliminate span condition. (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 )

**Assertion**
Ref	Expression
lcfrlem41	⊢ (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 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) 𝑊(𝑓,𝑔)

**Proof of Theorem lcfrlem41**
Step	Hyp	Ref	Expression
1		lcfrlem38.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem38.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfrlem38.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfrlem38.p	. . 3 ⊢ + = (+_g‘𝑈)
5		eqid 2737	. . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
6		lcfrlem38.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
7		lcfrlem38.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
8		lcfrlem38.q	. . 3 ⊢ 𝑄 = (LSubSp‘𝐷)
9		lcfrlem38.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	9	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		lcfrlem38.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑄)
12	11	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → 𝐺 ∈ 𝑄)
13		lcfrlem38.e	. . 3 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
14		lcfrlem38.xe	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
15	14	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → 𝑋 ∈ 𝐸)
16		lcfrlem38.ye	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
17	16	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → 𝑌 ∈ 𝐸)
18		simpr 488	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌}))
19	1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 15, 17, 18	lcfrlem6 39298	. 2 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) = ((LSpan‘𝑈)‘{𝑌})) → (𝑋 + 𝑌) ∈ 𝐸)
20		lcfrlem38.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
21		lcfrlem38.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
22	9	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	11	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝐺 ∈ 𝑄)
24		lcfrlem38.gs	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
25	24	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝐺 ⊆ 𝐶)
26	14	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝑋 ∈ 𝐸)
27	16	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝑌 ∈ 𝐸)
28		lcfrlem38.z	. . 3 ⊢ 0 = (0_g‘𝑈)
29		lcfrlem38.x	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 )
30	29	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝑋 ≠ 0 )
31		lcfrlem38.y	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 0 )
32	31	adantr 484	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → 𝑌 ≠ 0 )
33		simpr 488	. . 3 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌}))
34	1, 2, 3, 4, 20, 6, 7, 8, 21, 13, 22, 23, 25, 26, 27, 28, 30, 32, 5, 33	lcfrlem40 39333	. 2 ⊢ ((𝜑 ∧ ((LSpan‘𝑈)‘{𝑋}) ≠ ((LSpan‘𝑈)‘{𝑌})) → (𝑋 + 𝑌) ∈ 𝐸)
35	19, 34	pm2.61dane 3029	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 {crab 3065 ⊆ wss 3866 {csn 4541 ∪ ciun 4904 ‘cfv 6380 (class class class)co 7213 +_gcplusg 16802 0_gc0g 16944 LSubSpclss 19968 LSpanclspn 20008 LFnlclfn 36808 LKerclk 36836 LDualcld 36874 HLchlt 37101 LHypclh 37735 DVecHcdvh 38829 ocHcoch 39098

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-riotaBAD 36704

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-undef 8015 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-0g 16946 df-mre 17089 df-mrc 17090 df-acs 17092 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cntz 18711 df-oppg 18738 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lvec 20140 df-lsatoms 36727 df-lshyp 36728 df-lcv 36770 df-lfl 36809 df-lkr 36837 df-ldual 36875 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-llines 37249 df-lplanes 37250 df-lvols 37251 df-lines 37252 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-lhyp 37739 df-laut 37740 df-ldil 37855 df-ltrn 37856 df-trl 37910 df-tgrp 38494 df-tendo 38506 df-edring 38508 df-dveca 38754 df-disoa 38780 df-dvech 38830 df-dib 38890 df-dic 38924 df-dih 38980 df-doch 39099 df-djh 39146

This theorem is referenced by: lcfrlem42 39335

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