lcfrlem20 - Mathbox for Norm Megill

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Theorem lcfrlem20 42022

Description: Lemma for lcfr 42045. (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}))
lcfrlem20.e	⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))

**Assertion**
Ref	Expression
lcfrlem20	⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

**Proof of Theorem lcfrlem20**
Step	Hyp	Ref	Expression
1		lcfrlem17.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
2		lcfrlem17.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
3		eqid 2737	. . . 4 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
4		lcfrlem17.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		lcfrlem17.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		lcfrlem17.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	4, 5, 6	dvhlmod 41570	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		lcfrlem17.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3902	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10		lcfrlem17.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3902	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12	1, 2, 3, 7, 9, 11	lsmpr 21076	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
13	12	ineq1d 4160	. 2 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = (((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
14		eqid 2737	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
15		eqid 2737	. . 3 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
16		lcfrlem17.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑈)
17	4, 5, 6	dvhlvec 41569	. . 3 ⊢ (𝜑 → 𝑈 ∈ LVec)
18		lcfrlem17.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
19		lcfrlem17.z	. . . 4 ⊢ 0 = (0_g‘𝑈)
20		lcfrlem17.p	. . . . 5 ⊢ + = (+_g‘𝑈)
21		lcfrlem17.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
22	4, 18, 5, 1, 20, 19, 2, 16, 6, 8, 10, 21	lcfrlem17 42019	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
23	4, 18, 5, 1, 19, 15, 6, 22	dochsnshp 41913	. . 3 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSHyp‘𝑈))
24	1, 2, 19, 16, 7, 8	lsatlspsn 39453	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)
25	1, 2, 19, 16, 7, 10	lsatlspsn 39453	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
26		lcfrlem20.e	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))
27	1, 20	lmodvacl 20861	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
28	7, 9, 11, 27	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
29	28	snssd 4753	. . . . . 6 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
30	4, 5, 1, 14, 18	dochlss 41814	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
31	6, 29, 30	syl2anc 585	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
32	1, 14, 2, 7, 31, 9	ellspsn5b 20981	. . . 4 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ↔ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{(𝑋 + 𝑌)})))
33	26, 32	mtbid 324	. . 3 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{(𝑋 + 𝑌)}))
34	14, 3, 15, 16, 17, 23, 24, 25, 21, 33	lshpat 39516	. 2 ⊢ (𝜑 → (((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
35	13, 34	eqeltrd 2837	1 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ∩ cin 3889 ⊆ wss 3890 {csn 4568 {cpr 4570 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 +_gcplusg 17211 0_gc0g 17393 LSSumclsm 19600 LModclmod 20846 LSubSpclss 20917 LSpanclspn 20957 LSAtomsclsa 39434 LSHypclsh 39435 HLchlt 39810 LHypclh 40444 DVecHcdvh 41538 ocHcoch 41807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-riotaBAD 39413

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 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 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8169 df-undef 8216 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cntz 19283 df-oppg 19312 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-drng 20699 df-lmod 20848 df-lss 20918 df-lsp 20958 df-lvec 21090 df-lsatoms 39436 df-lshyp 39437 df-lcv 39479 df-oposet 39636 df-ol 39638 df-oml 39639 df-covers 39726 df-ats 39727 df-atl 39758 df-cvlat 39782 df-hlat 39811 df-llines 39958 df-lplanes 39959 df-lvols 39960 df-lines 39961 df-psubsp 39963 df-pmap 39964 df-padd 40256 df-lhyp 40448 df-laut 40449 df-ldil 40564 df-ltrn 40565 df-trl 40619 df-tgrp 41203 df-tendo 41215 df-edring 41217 df-dveca 41463 df-disoa 41489 df-dvech 41539 df-dib 41599 df-dic 41633 df-dih 41689 df-doch 41808 df-djh 41855

This theorem is referenced by: lcfrlem21 42023

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