lclkrlem2i - Mathbox for Norm Megill

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Theorem lclkrlem2i 42540

Description: Lemma for lclkr 42558. 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 42536	. 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 42539	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
44	21, 43	pm2.61dane 3043	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 2956 ∖ cdif 3896 {csn 4584 ‘cfv 6531 (class class class)co 7412 Basecbs 17367 +_gcplusg 17408 Scalarcsca 17411 0_gc0g 17590 LSSumclsm 19828 LSpanclspn 21226 LSHypclsh 40000 LFnlclfn 40082 LKerclk 40110 LDualcld 40148 HLchlt 40375 LHypclh 41009 DVecHcdvh 42103 ocHcoch 42372

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 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-riotaBAD 39978

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8227 df-undef 8274 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-n0 12588 df-z 12675 df-uz 12947 df-fz 13621 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-sca 17424 df-vsca 17425 df-0g 17592 df-mre 17736 df-mrc 17737 df-acs 17739 df-proset 18448 df-poset 18467 df-plt 18482 df-lub 18498 df-glb 18499 df-join 18500 df-meet 18501 df-p0 18577 df-p1 18578 df-lat 18586 df-clat 18653 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-subg 19313 df-cntz 19511 df-oppg 19540 df-lsm 19830 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-invr 20598 df-dvr 20611 df-drng 20962 df-lmod 21117 df-lss 21187 df-lsp 21227 df-lvec 21358 df-lsatoms 40001 df-lshyp 40002 df-lcv 40044 df-lfl 40083 df-lkr 40111 df-ldual 40149 df-oposet 40201 df-ol 40203 df-oml 40204 df-covers 40291 df-ats 40292 df-atl 40323 df-cvlat 40347 df-hlat 40376 df-llines 40523 df-lplanes 40524 df-lvols 40525 df-lines 40526 df-psubsp 40528 df-pmap 40529 df-padd 40821 df-lhyp 41013 df-laut 41014 df-ldil 41129 df-ltrn 41130 df-trl 41184 df-tgrp 41768 df-tendo 41780 df-edring 41782 df-dveca 42028 df-disoa 42054 df-dvech 42104 df-dib 42164 df-dic 42198 df-dih 42254 df-doch 42373 df-djh 42420

This theorem is used by: lclkrlem2l 42543

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