lcfrlem21 - Mathbox for Norm Megill

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Theorem lcfrlem21 40890

Description: Lemma for lcfr 40912. (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 40889	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
19	1, 3, 9	dvhlmod 40437	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
20	11	eldifad 3952	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21	13	eldifad 3952	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
22	4, 5	lmodcom 20743	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
23	19, 20, 21, 22	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
24	23	sneqd 4632	. . . . . . 7 ⊢ (𝜑 → {(𝑋 + 𝑌)} = {(𝑌 + 𝑋)})
25	24	fveq2d 6885	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = ( ⊥ ‘{(𝑌 + 𝑋)}))
26	25	eleq2d 2811	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ↔ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})))
27	26	biimprd 247	. . . 4 ⊢ (𝜑 → (𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}) → 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
28	27	con3dimp 408	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) → ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)}))
29		prcom 4728	. . . . . . . 8 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋}
30	29	fveq2i 6884	. . . . . . 7 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋})
31	30	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋}))
32	31, 25	ineq12d 4205	. . . . 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 2988	. . . . . 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 40889	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑌, 𝑋}) ∩ ( ⊥ ‘{(𝑌 + 𝑋)})) ∈ 𝐴)
41	33, 40	eqeltrd 2825	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑌 + 𝑋)})) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
42	28, 41	syldan 590	. 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 40888	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
44	18, 42, 43	mpjaodan 955	1 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3937 ∩ cin 3939 {csn 4620 {cpr 4622 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 +_gcplusg 17195 0_gc0g 17383 LModclmod 20695 LSpanclspn 20807 LSAtomsclsa 38300 HLchlt 38676 LHypclh 39311 DVecHcdvh 40405 ocHcoch 40674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-riotaBAD 38279

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-sca 17211 df-vsca 17212 df-0g 17385 df-mre 17528 df-mrc 17529 df-acs 17531 df-proset 18249 df-poset 18267 df-plt 18284 df-lub 18300 df-glb 18301 df-join 18302 df-meet 18303 df-p0 18379 df-p1 18380 df-lat 18386 df-clat 18453 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19039 df-cntz 19222 df-oppg 19251 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20578 df-lmod 20697 df-lss 20768 df-lsp 20808 df-lvec 20940 df-lsatoms 38302 df-lshyp 38303 df-lcv 38345 df-oposet 38502 df-ol 38504 df-oml 38505 df-covers 38592 df-ats 38593 df-atl 38624 df-cvlat 38648 df-hlat 38677 df-llines 38825 df-lplanes 38826 df-lvols 38827 df-lines 38828 df-psubsp 38830 df-pmap 38831 df-padd 39123 df-lhyp 39315 df-laut 39316 df-ldil 39431 df-ltrn 39432 df-trl 39486 df-tgrp 40070 df-tendo 40082 df-edring 40084 df-dveca 40330 df-disoa 40356 df-dvech 40406 df-dib 40466 df-dic 40500 df-dih 40556 df-doch 40675 df-djh 40722

This theorem is referenced by: lcfrlem22 40891 lcfrlem40 40909

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