lcfrlem21 - Mathbox for Norm Megill

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Theorem lcfrlem21 42199

Description: Lemma for lcfr 42221. (Contributed by NM, 11-Mar-2015.)

**Hypotheses**
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 = (0_g‘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}))

**Assertion**
Ref	Expression
lcfrlem21	⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

**Proof of Theorem lcfrlem21**
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 = (0_g‘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 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		lcfrlem17.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	11	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
13		lcfrlem17.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
15		lcfrlem17.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
16	15	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
17		simpr 489	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))
18	1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17	lcfrlem20 42198	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
19	1, 3, 9	dvhlmod 41746	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
20	11	eldifad 3919	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21	13	eldifad 3919	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
22	4, 5	lmodcom 20998	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
23	19, 20, 21, 22	syl3anc 1394	. . . . . . . 8 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
24	23	sneqd 4597	. . . . . . 7 ⊢ (𝜑 → {(𝑋 + 𝑌)} = {(𝑌 + 𝑋)})
25	24	fveq2d 6875	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = ( ⊥ ‘{(𝑌 + 𝑋)}))
26	25	eleq2d 2851	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ↔ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})))
27	26	biimprd 251	. . . 4 ⊢ (𝜑 → (𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}) → 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
28	27	con3dimp 413	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}))
29		prcom 4694	. . . . . . . 8 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋}
30	29	fveq2i 6874	. . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋})
31	30	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋}))
32	31, 25	ineq12d 4176	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)})))
33	32	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)})))
34	9	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
35	13	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
36	11	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
37	15	necomd 3015	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋}))
38	37	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋}))
39		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}))
40	1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 38, 39	lcfrlem20 42198	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)})) ∈ 𝐴)
41	33, 40	eqeltrd 2865	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
42	28, 41	syldan 602	. 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 42197	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
44	18, 42, 43	mpjaodan 973	1 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∖ cdif 3904 ∩ cin 3906 {csn 4585 {cpr 4587 ‘cfv 6525 (class class class)co 7400 Basecbs 17259 +_gcplusg 17300 0_gc0g 17482 LModclmod 20950 LSpanclspn 21061 LSAtomsclsa 39610 HLchlt 39986 LHypclh 40620 DVecHcdvh 41714 ocHcoch 41983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-riotaBAD 39589

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-tpos 8210 df-undef 8257 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-sca 17316 df-vsca 17317 df-0g 17484 df-mre 17628 df-mrc 17629 df-acs 17631 df-proset 18340 df-poset 18359 df-plt 18374 df-lub 18390 df-glb 18391 df-join 18392 df-meet 18393 df-p0 18469 df-p1 18470 df-lat 18478 df-clat 18545 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-grp 18993 df-minusg 18994 df-sbg 18995 df-subg 19180 df-cntz 19378 df-oppg 19407 df-lsm 19697 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-oppr 20410 df-dvdsr 20430 df-unit 20431 df-invr 20461 df-dvr 20474 df-drng 20806 df-lmod 20952 df-lss 21022 df-lsp 21062 df-lvec 21193 df-lsatoms 39612 df-lshyp 39613 df-lcv 39655 df-oposet 39812 df-ol 39814 df-oml 39815 df-covers 39902 df-ats 39903 df-atl 39934 df-cvlat 39958 df-hlat 39987 df-llines 40134 df-lplanes 40135 df-lvols 40136 df-lines 40137 df-psubsp 40139 df-pmap 40140 df-padd 40432 df-lhyp 40624 df-laut 40625 df-ldil 40740 df-ltrn 40741 df-trl 40795 df-tgrp 41379 df-tendo 41391 df-edring 41393 df-dveca 41639 df-disoa 41665 df-dvech 41715 df-dib 41775 df-dic 41809 df-dih 41865 df-doch 41984 df-djh 42031

This theorem is referenced by: lcfrlem22 42200 lcfrlem40 42218

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