lclkrlem2i - Mathbox for Norm Megill

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Theorem lclkrlem2i 42396

Description: Lemma for lclkr 42414. Eliminate the (𝐿‘𝐸) ≠ (𝐿‘𝐺) hypothesis. (Contributed by NM, 17-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})))
lclkrlem2i.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lclkrlem2i.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))

**Assertion**
Ref	Expression
lclkrlem2i	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

**Proof of Theorem lclkrlem2i**
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.z	. . 3 ⊢ 0 = (0_g‘𝑈)
6		lclkrlem2f.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
7		lclkrlem2f.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
8		lclkrlem2f.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
9		lclkrlem2f.p	. . 3 ⊢ + = (+_g‘𝐷)
10		lclkrlem2f.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		lclkrlem2i.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → 𝑋 ∈ (𝑉 ∖ { 0 }))
14		lclkrlem2f.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
15	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → 𝐸 ∈ 𝐹)
16		lclkrlem2f.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
17	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → 𝐺 ∈ 𝐹)
18		lclkrlem2f.le	. . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
19	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
20		simpr 490	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → (𝐿‘𝐸) = (𝐿‘𝐺))
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20	lclkrlem2e 42392	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
22		lclkrlem2f.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
23		lclkrlem2f.q	. . 3 ⊢ 𝑄 = (0_g‘𝑆)
24		lclkrlem2f.a	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
25		lclkrlem2f.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
26		lclkrlem2f.j	. . 3 ⊢ 𝐽 = (LSHyp‘𝑈)
27	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28		lclkrlem2f.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
29	28	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝐵 ∈ (𝑉 ∖ { 0 }))
30	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝐸 ∈ 𝐹)
31	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝐺 ∈ 𝐹)
32	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
33		lclkrlem2f.lg	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
34	33	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
35		lclkrlem2f.kb	. . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
36	35	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
37		lclkrlem2f.nx	. . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
38	37	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
39	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝑋 ∈ (𝑉 ∖ { 0 }))
40		lclkrlem2i.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
41	40	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝑌 ∈ (𝑉 ∖ { 0 }))
42		simpr 490	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
43	1, 2, 3, 4, 22, 23, 5, 24, 25, 6, 26, 7, 8, 9, 27, 29, 30, 31, 32, 34, 36, 38, 39, 41, 42	lclkrlem2h 42395	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
44	21, 43	pm2.61dane 3044	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 1570 ∈ wcel 2145 ≠ wne 2957 ∖ cdif 3899 {csn 4587 ‘cfv 6537 (class class class)co 7417 Basecbs 17307 +_gcplusg 17348 Scalarcsca 17351 0_gc0g 17530 LSSumclsm 19767 LSpanclspn 21161 LSHypclsh 39856 LFnlclfn 39938 LKerclk 39966 LDualcld 40004 HLchlt 40231 LHypclh 40865 DVecHcdvh 41959 ocHcoch 42228

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-riotaBAD 39834

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-n0 12533 df-z 12620 df-uz 12892 df-fz 13566 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-sca 17364 df-vsca 17365 df-0g 17532 df-mre 17676 df-mrc 17677 df-acs 17679 df-proset 18388 df-poset 18407 df-plt 18422 df-lub 18438 df-glb 18439 df-join 18440 df-meet 18441 df-p0 18517 df-p1 18518 df-lat 18526 df-clat 18593 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-grp 19066 df-minusg 19067 df-sbg 19068 df-subg 19252 df-cntz 19450 df-oppg 19479 df-lsm 19769 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-invr 20535 df-dvr 20548 df-drng 20898 df-lmod 21052 df-lss 21122 df-lsp 21162 df-lvec 21293 df-lsatoms 39857 df-lshyp 39858 df-lcv 39900 df-lfl 39939 df-lkr 39967 df-ldual 40005 df-oposet 40057 df-ol 40059 df-oml 40060 df-covers 40147 df-ats 40148 df-atl 40179 df-cvlat 40203 df-hlat 40232 df-llines 40379 df-lplanes 40380 df-lvols 40381 df-lines 40382 df-psubsp 40384 df-pmap 40385 df-padd 40677 df-lhyp 40869 df-laut 40870 df-ldil 40985 df-ltrn 40986 df-trl 41040 df-tgrp 41624 df-tendo 41636 df-edring 41638 df-dveca 41884 df-disoa 41910 df-dvech 41960 df-dib 42020 df-dic 42054 df-dih 42110 df-doch 42229 df-djh 42276

This theorem is used by: lclkrlem2l 42399

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