lcfrlem21 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem21		Structured version Visualization version GIF version

Theorem lcfrlem21 42315

Description: Lemma for lcfr 42337. (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 42314	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
19	1, 3, 9	dvhlmod 41862	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
20	11	eldifad 3918	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21	13	eldifad 3918	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
22	4, 5	lmodcom 21010	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
23	19, 20, 21, 22	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
24	23	sneqd 4602	. . . . . . 7 ⊢ (𝜑 → {(𝑋 + 𝑌)} = {(𝑌 + 𝑋)})
25	24	fveq2d 6887	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = ( ⊥ ‘{(𝑌 + 𝑋)}))
26	25	eleq2d 2849	. . . . 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 4699	. . . . . . . 8 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋}
30	29	fveq2i 6886	. . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋})
31	30	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋}))
32	31, 25	ineq12d 4175	. . . . 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 3013	. . . . . 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 42314	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)})) ∈ 𝐴)
41	33, 40	eqeltrd 2863	. . 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 42313	. 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 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3903 ∩ cin 3905 {csn 4590 {cpr 4592 ‘cfv 6538 (class class class)co 7412 Basecbs 17270 +_gcplusg 17311 0_gc0g 17493 LModclmod 20962 LSpanclspn 21073 LSAtomsclsa 39726 HLchlt 40102 LHypclh 40736 DVecHcdvh 41830 ocHcoch 42099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-riotaBAD 39705

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8223 df-undef 8270 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-n0 12506 df-z 12593 df-uz 12864 df-fz 13537 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17495 df-mre 17639 df-mrc 17640 df-acs 17642 df-proset 18351 df-poset 18370 df-plt 18385 df-lub 18401 df-glb 18402 df-join 18403 df-meet 18404 df-p0 18480 df-p1 18481 df-lat 18489 df-clat 18556 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-grp 19004 df-minusg 19005 df-sbg 19006 df-subg 19190 df-cntz 19388 df-oppg 19417 df-lsm 19707 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-oppr 20420 df-dvdsr 20440 df-unit 20441 df-invr 20471 df-dvr 20484 df-drng 20816 df-lmod 20964 df-lss 21034 df-lsp 21074 df-lvec 21205 df-lsatoms 39728 df-lshyp 39729 df-lcv 39771 df-oposet 39928 df-ol 39930 df-oml 39931 df-covers 40018 df-ats 40019 df-atl 40050 df-cvlat 40074 df-hlat 40103 df-llines 40250 df-lplanes 40251 df-lvols 40252 df-lines 40253 df-psubsp 40255 df-pmap 40256 df-padd 40548 df-lhyp 40740 df-laut 40741 df-ldil 40856 df-ltrn 40857 df-trl 40911 df-tgrp 41495 df-tendo 41507 df-edring 41509 df-dveca 41755 df-disoa 41781 df-dvech 41831 df-dib 41891 df-dic 41925 df-dih 41981 df-doch 42100 df-djh 42147

This theorem is referenced by: lcfrlem22 42316 lcfrlem40 42334

W3C validator