lcfrlem21 - Mathbox for Norm Megill

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Theorem lcfrlem21 41819

Description: Lemma for lcfr 41841. (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 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		lcfrlem17.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
13		lcfrlem17.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
15		lcfrlem17.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
16	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
17		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))
18	1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17	lcfrlem20 41818	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
19	1, 3, 9	dvhlmod 41366	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
20	11	eldifad 3913	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21	13	eldifad 3913	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
22	4, 5	lmodcom 20859	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
23	19, 20, 21, 22	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
24	23	sneqd 4592	. . . . . . 7 ⊢ (𝜑 → {(𝑋 + 𝑌)} = {(𝑌 + 𝑋)})
25	24	fveq2d 6838	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = ( ⊥ ‘{(𝑌 + 𝑋)}))
26	25	eleq2d 2822	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ↔ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})))
27	26	biimprd 248	. . . 4 ⊢ (𝜑 → (𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}) → 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
28	27	con3dimp 408	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}))
29		prcom 4689	. . . . . . . 8 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋}
30	29	fveq2i 6837	. . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋})
31	30	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋}))
32	31, 25	ineq12d 4173	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)})))
33	32	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)})))
34	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
35	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → 𝑌 ∈ (𝑉 ∖ { 0 }))
36	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → 𝑋 ∈ (𝑉 ∖ { 0 }))
37	15	necomd 2987	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋}))
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋}))
39		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}))
40	1, 2, 3, 4, 5, 6, 7, 8, 34, 35, 36, 38, 39	lcfrlem20 41818	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)})) ∈ 𝐴)
41	33, 40	eqeltrd 2836	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
42	28, 41	syldan 591	. 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 41817	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
44	18, 42, 43	mpjaodan 960	1 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 ∩ cin 3900 {csn 4580 {cpr 4582 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 +_gcplusg 17177 0_gc0g 17359 LModclmod 20811 LSpanclspn 20922 LSAtomsclsa 39230 HLchlt 39606 LHypclh 40240 DVecHcdvh 41334 ocHcoch 41603

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-riotaBAD 39209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-0g 17361 df-mre 17505 df-mrc 17506 df-acs 17508 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cntz 19246 df-oppg 19275 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-drng 20664 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lvec 21055 df-lsatoms 39232 df-lshyp 39233 df-lcv 39275 df-oposet 39432 df-ol 39434 df-oml 39435 df-covers 39522 df-ats 39523 df-atl 39554 df-cvlat 39578 df-hlat 39607 df-llines 39754 df-lplanes 39755 df-lvols 39756 df-lines 39757 df-psubsp 39759 df-pmap 39760 df-padd 40052 df-lhyp 40244 df-laut 40245 df-ldil 40360 df-ltrn 40361 df-trl 40415 df-tgrp 40999 df-tendo 41011 df-edring 41013 df-dveca 41259 df-disoa 41285 df-dvech 41335 df-dib 41395 df-dic 41429 df-dih 41485 df-doch 41604 df-djh 41651

This theorem is referenced by: lcfrlem22 41820 lcfrlem40 41838

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