lcfrlem19 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem19		Structured version Visualization version GIF version

Theorem lcfrlem19 42060

Description: Lemma for lcfr 42084. (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
lcfrlem19	⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))

**Proof of Theorem lcfrlem19**
Step	Hyp	Ref	Expression
1		lcfrlem17.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem17.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfrlem17.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfrlem17.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		lcfrlem17.z	. . . 4 ⊢ 0 = (0_g‘𝑈)
6		lcfrlem17.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		lcfrlem17.p	. . . . 5 ⊢ + = (+_g‘𝑈)
8		lcfrlem17.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
9		lcfrlem17.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈)
10		lcfrlem17.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11		lcfrlem17.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
12		lcfrlem17.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
13	1, 2, 3, 4, 7, 5, 8, 9, 6, 10, 11, 12	lcfrlem17 42058	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
14	1, 2, 3, 4, 5, 6, 13	dochnel 41892	. . 3 ⊢ (𝜑 → ¬ (𝑋 + 𝑌) ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))
15	1, 3, 6	dvhlmod 41609	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
16	15	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∧ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))) → 𝑈 ∈ LMod)
17	10	eldifad 3902	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
18	11	eldifad 3902	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19	4, 7	lmodvacl 20872	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
20	15, 17, 18, 19	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
21	20	snssd 4725	. . . . . 6 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
22		eqid 2740	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
23	1, 3, 4, 22, 2	dochlss 41853	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
24	6, 21, 23	syl2anc 590	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
25	24	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∧ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
26		simpr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∧ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))) → (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∧ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
27	7, 22	lssvacl 20940	. . . 4 ⊢ (((𝑈 ∈ LMod ∧ ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈)) ∧ (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∧ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))
28	16, 25, 26, 27	syl21anc 843	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∧ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))
29	14, 28	mtand 821	. 2 ⊢ (𝜑 → ¬ (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∧ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
30		ianor 989	. 2 ⊢ (¬ (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∧ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) ↔ (¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
31	29, 30	sylib 219	1 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∖ cdif 3887 ⊆ wss 3890 {csn 4562 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 +_gcplusg 17218 0_gc0g 17400 LModclmod 20857 LSubSpclss 20928 LSpanclspn 20968 LSAtomsclsa 39473 HLchlt 39849 LHypclh 40483 DVecHcdvh 41577 ocHcoch 41846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-riotaBAD 39452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-tpos 8173 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17402 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cntz 19290 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-lsatoms 39475 df-oposet 39675 df-ol 39677 df-oml 39678 df-covers 39765 df-ats 39766 df-atl 39797 df-cvlat 39821 df-hlat 39850 df-llines 39997 df-lplanes 39998 df-lvols 39999 df-lines 40000 df-psubsp 40002 df-pmap 40003 df-padd 40295 df-lhyp 40487 df-laut 40488 df-ldil 40603 df-ltrn 40604 df-trl 40658 df-tendo 41254 df-edring 41256 df-disoa 41528 df-dvech 41578 df-dib 41638 df-dic 41672 df-dih 41728 df-doch 41847

This theorem is referenced by: lcfrlem21 42062

W3C validator