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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2i | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38702. Eliminate the (𝐿‘𝐸) ≠ (𝐿‘𝐺) hypothesis. (Contributed by NM, 17-Jan-2015.) |
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 = (0g‘𝑆) |
lclkrlem2f.z | ⊢ 0 = (0g‘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 })) |
Ref | Expression |
---|---|
lclkrlem2i | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
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 = (0g‘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 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
12 | lclkrlem2i.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
13 | 12 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
14 | lclkrlem2f.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
15 | 14 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → 𝐸 ∈ 𝐹) |
16 | lclkrlem2f.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
17 | 16 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → 𝐺 ∈ 𝐹) |
18 | lclkrlem2f.le | . . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
19 | 18 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
20 | simpr 487 | . . 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 38680 | . 2 ⊢ ((𝜑 ∧ (𝐿‘𝐸) = (𝐿‘𝐺)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
22 | lclkrlem2f.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑈) | |
23 | lclkrlem2f.q | . . 3 ⊢ 𝑄 = (0g‘𝑆) | |
24 | lclkrlem2f.a | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
25 | lclkrlem2f.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
26 | lclkrlem2f.j | . . 3 ⊢ 𝐽 = (LSHyp‘𝑈) | |
27 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
28 | lclkrlem2f.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) | |
29 | 28 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝐵 ∈ (𝑉 ∖ { 0 })) |
30 | 14 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝐸 ∈ 𝐹) |
31 | 16 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝐺 ∈ 𝐹) |
32 | 18 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
33 | lclkrlem2f.lg | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
34 | 33 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
35 | lclkrlem2f.kb | . . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄) | |
36 | 35 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → ((𝐸 + 𝐺)‘𝐵) = 𝑄) |
37 | lclkrlem2f.nx | . . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) | |
38 | 37 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) |
39 | 12 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
40 | lclkrlem2i.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
41 | 40 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
42 | simpr 487 | . . 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 38683 | . 2 ⊢ ((𝜑 ∧ (𝐿‘𝐸) ≠ (𝐿‘𝐺)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
44 | 21, 43 | pm2.61dane 3103 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∖ cdif 3926 {csn 4560 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 +gcplusg 16560 Scalarcsca 16563 0gc0g 16708 LSSumclsm 18754 LSpanclspn 19738 LSHypclsh 36144 LFnlclfn 36226 LKerclk 36254 LDualcld 36292 HLchlt 36519 LHypclh 37153 DVecHcdvh 38247 ocHcoch 38516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-riotaBAD 36122 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-undef 7932 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-0g 16710 df-mre 16852 df-mrc 16853 df-acs 16855 df-proset 17533 df-poset 17551 df-plt 17563 df-lub 17579 df-glb 17580 df-join 17581 df-meet 17582 df-p0 17644 df-p1 17645 df-lat 17651 df-clat 17713 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-submnd 17952 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-cntz 18442 df-oppg 18469 df-lsm 18756 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-ring 19294 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-invr 19417 df-dvr 19428 df-drng 19499 df-lmod 19631 df-lss 19699 df-lsp 19739 df-lvec 19870 df-lsatoms 36145 df-lshyp 36146 df-lcv 36188 df-lfl 36227 df-lkr 36255 df-ldual 36293 df-oposet 36345 df-ol 36347 df-oml 36348 df-covers 36435 df-ats 36436 df-atl 36467 df-cvlat 36491 df-hlat 36520 df-llines 36667 df-lplanes 36668 df-lvols 36669 df-lines 36670 df-psubsp 36672 df-pmap 36673 df-padd 36965 df-lhyp 37157 df-laut 37158 df-ldil 37273 df-ltrn 37274 df-trl 37328 df-tgrp 37912 df-tendo 37924 df-edring 37926 df-dveca 38172 df-disoa 38198 df-dvech 38248 df-dib 38308 df-dic 38342 df-dih 38398 df-doch 38517 df-djh 38564 |
This theorem is referenced by: lclkrlem2l 38687 |
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