lclkrlem2h - Mathbox for Norm Megill

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Theorem lclkrlem2h 42711

Description: Lemma for lclkr 42730. Eliminate the (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 hypothesis. (Contributed by NM, 16-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrlem2f.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem2f.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem2f.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem2f.v	⊢ 𝑉 = (Base‘𝑈)
lclkrlem2f.s	⊢ 𝑆 = (Scalar‘𝑈)
lclkrlem2f.q	⊢ 𝑄 = (0_g‘𝑆)
lclkrlem2f.z	⊢ 0 = (0_g‘𝑈)
lclkrlem2f.a	⊢ ⊕ = (LSSum‘𝑈)
lclkrlem2f.n	⊢ 𝑁 = (LSpan‘𝑈)
lclkrlem2f.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrlem2f.j	⊢ 𝐽 = (LSHyp‘𝑈)
lclkrlem2f.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrlem2f.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrlem2f.p	⊢ + = (+_g‘𝐷)
lclkrlem2f.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem2f.b	⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
lclkrlem2f.e	⊢ (𝜑 → 𝐸 ∈ 𝐹)
lclkrlem2f.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lclkrlem2f.le	⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
lclkrlem2f.lg	⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
lclkrlem2f.kb	⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
lclkrlem2f.nx	⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
lclkrlem2h.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lclkrlem2h.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lclkrlem2h.ne	⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))

**Assertion**
Ref	Expression
lclkrlem2h	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

**Proof of Theorem lclkrlem2h**
Step	Hyp	Ref	Expression
1		lclkrlem2f.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lclkrlem2f.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lclkrlem2f.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lclkrlem2f.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		lclkrlem2f.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
6		lclkrlem2f.q	. . 3 ⊢ 𝑄 = (0_g‘𝑆)
7		lclkrlem2f.z	. . 3 ⊢ 0 = (0_g‘𝑈)
8		lclkrlem2f.a	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
9		lclkrlem2f.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
10		lclkrlem2f.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
11		lclkrlem2f.j	. . 3 ⊢ 𝐽 = (LSHyp‘𝑈)
12		lclkrlem2f.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
13		lclkrlem2f.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
14		lclkrlem2f.p	. . 3 ⊢ + = (+_g‘𝐷)
15		lclkrlem2f.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16	15	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		lclkrlem2f.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
18	17	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐵 ∈ (𝑉 ∖ { 0 }))
19		lclkrlem2f.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
20	19	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐸 ∈ 𝐹)
21		lclkrlem2f.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
22	21	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐺 ∈ 𝐹)
23		lclkrlem2f.le	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
24	23	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
25		lclkrlem2f.lg	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
26	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
27		lclkrlem2f.kb	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
28	27	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
29		lclkrlem2f.nx	. . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
30	29	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
31		lclkrlem2h.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
32	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑋 ∈ (𝑉 ∖ { 0 }))
33		lclkrlem2h.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
34	33	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑌 ∈ (𝑉 ∖ { 0 }))
35		lclkrlem2h.ne	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
36	35	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
37		simpr 490	. . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)
38	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 37	lclkrlem2g 42710	. 2 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
39	1, 3, 2, 4, 15	dochoc1 42558	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
40	39	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
41	1, 3, 15	dvhlvec 42306	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
42	1, 3, 15	dvhlmod 42307	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
43	10, 13, 14, 42, 19, 21	ldualvaddcl 40327	. . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
44	4, 11, 10, 12, 41, 43	lkrshpor 40304	. . . . . 6 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
45	44	orcanai 1018	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
46	45	fveq2d 6897	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
47	46	fveq2d 6897	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
48	40, 47, 45	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
49	38, 48	pm2.61dan 825	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2146 ≠ wne 2958 ∖ cdif 4047 {csn 4587 ‘cfv 6546 (class class class)co 7427 Basecbs 17399 +_gcplusg 17440 Scalarcsca 17443 0_gc0g 17622 LSSumclsm 19860 LSpanclspn 21258 LSHypclsh 40172 LFnlclfn 40254 LKerclk 40282 LDualcld 40320 HLchlt 40547 LHypclh 41181 DVecHcdvh 42275 ocHcoch 42544

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-riotaBAD 40150

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 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 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7700 df-om 7885 df-1st 8008 df-2nd 8009 df-tpos 8254 df-undef 8301 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-er 8728 df-map 8860 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-n0 12617 df-z 12704 df-uz 12976 df-fz 13650 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-sca 17456 df-vsca 17457 df-0g 17624 df-mre 17768 df-mrc 17769 df-acs 17771 df-proset 18480 df-poset 18499 df-plt 18514 df-lub 18530 df-glb 18531 df-join 18532 df-meet 18533 df-p0 18609 df-p1 18610 df-lat 18618 df-clat 18685 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-submnd 18991 df-grp 19159 df-minusg 19160 df-sbg 19161 df-subg 19345 df-cntz 19543 df-oppg 19572 df-lsm 19862 df-cmn 20008 df-abl 20009 df-mgp 20373 df-rng 20387 df-ur 20420 df-ring 20473 df-oppr 20579 df-dvdsr 20599 df-unit 20600 df-invr 20630 df-dvr 20643 df-drng 20994 df-lmod 21149 df-lss 21219 df-lsp 21259 df-lvec 21390 df-lsatoms 40173 df-lshyp 40174 df-lcv 40216 df-lfl 40255 df-lkr 40283 df-ldual 40321 df-oposet 40373 df-ol 40375 df-oml 40376 df-covers 40463 df-ats 40464 df-atl 40495 df-cvlat 40519 df-hlat 40548 df-llines 40695 df-lplanes 40696 df-lvols 40697 df-lines 40698 df-psubsp 40700 df-pmap 40701 df-padd 40993 df-lhyp 41185 df-laut 41186 df-ldil 41301 df-ltrn 41302 df-trl 41356 df-tgrp 41940 df-tendo 41952 df-edring 41954 df-dveca 42200 df-disoa 42226 df-dvech 42276 df-dib 42336 df-dic 42370 df-dih 42426 df-doch 42545 df-djh 42592

This theorem is used by: lclkrlem2i 42712

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