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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2h | Structured version Visualization version GIF version |
Description: Lemma for lclkr 39192. Eliminate the (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 hypothesis. (Contributed by NM, 16-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}))) |
lclkrlem2h.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2h.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2h.ne | ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺)) |
Ref | Expression |
---|---|
lclkrlem2h | ⊢ (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.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑈) | |
6 | lclkrlem2f.q | . . 3 ⊢ 𝑄 = (0g‘𝑆) | |
7 | lclkrlem2f.z | . . 3 ⊢ 0 = (0g‘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 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
17 | lclkrlem2f.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) | |
18 | 17 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐵 ∈ (𝑉 ∖ { 0 })) |
19 | lclkrlem2f.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
20 | 19 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐸 ∈ 𝐹) |
21 | lclkrlem2f.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
22 | 21 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝐺 ∈ 𝐹) |
23 | lclkrlem2f.le | . . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
24 | 23 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
25 | lclkrlem2f.lg | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
26 | 25 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
27 | lclkrlem2f.kb | . . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄) | |
28 | 27 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ((𝐸 + 𝐺)‘𝐵) = 𝑄) |
29 | lclkrlem2f.nx | . . . 4 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) | |
30 | 29 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) |
31 | lclkrlem2h.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
32 | 31 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
33 | lclkrlem2h.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
34 | 33 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
35 | lclkrlem2h.ne | . . . 4 ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺)) | |
36 | 35 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘𝐸) ≠ (𝐿‘𝐺)) |
37 | simpr 488 | . . 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 39172 | . 2 ⊢ ((𝜑 ∧ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
39 | 1, 3, 2, 4, 15 | dochoc1 39020 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) |
40 | 39 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) |
41 | 1, 3, 15 | dvhlvec 38768 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) |
42 | 1, 3, 15 | dvhlmod 38769 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
43 | 10, 13, 14, 42, 19, 21 | ldualvaddcl 36789 | . . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹) |
44 | 4, 11, 10, 12, 41, 43 | lkrshpor 36766 | . . . . . 6 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉)) |
45 | 44 | orcanai 1002 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → (𝐿‘(𝐸 + 𝐺)) = 𝑉) |
46 | 45 | fveq2d 6680 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉)) |
47 | 46 | fveq2d 6680 | . . 3 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉))) |
48 | 40, 47, 45 | 3eqtr4d 2783 | . 2 ⊢ ((𝜑 ∧ ¬ (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
49 | 38, 48 | pm2.61dan 813 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∖ cdif 3840 {csn 4516 ‘cfv 6339 (class class class)co 7172 Basecbs 16588 +gcplusg 16670 Scalarcsca 16673 0gc0g 16818 LSSumclsm 18879 LSpanclspn 19864 LSHypclsh 36634 LFnlclfn 36716 LKerclk 36744 LDualcld 36782 HLchlt 37009 LHypclh 37643 DVecHcdvh 38737 ocHcoch 39006 |
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 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-riotaBAD 36612 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-om 7602 df-1st 7716 df-2nd 7717 df-tpos 7923 df-undef 7970 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-map 8441 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-n0 11979 df-z 12065 df-uz 12327 df-fz 12984 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-mulr 16684 df-sca 16686 df-vsca 16687 df-0g 16820 df-mre 16962 df-mrc 16963 df-acs 16965 df-proset 17656 df-poset 17674 df-plt 17686 df-lub 17702 df-glb 17703 df-join 17704 df-meet 17705 df-p0 17767 df-p1 17768 df-lat 17774 df-clat 17836 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-submnd 18075 df-grp 18224 df-minusg 18225 df-sbg 18226 df-subg 18396 df-cntz 18567 df-oppg 18594 df-lsm 18881 df-cmn 19028 df-abl 19029 df-mgp 19361 df-ur 19373 df-ring 19420 df-oppr 19497 df-dvdsr 19515 df-unit 19516 df-invr 19546 df-dvr 19557 df-drng 19625 df-lmod 19757 df-lss 19825 df-lsp 19865 df-lvec 19996 df-lsatoms 36635 df-lshyp 36636 df-lcv 36678 df-lfl 36717 df-lkr 36745 df-ldual 36783 df-oposet 36835 df-ol 36837 df-oml 36838 df-covers 36925 df-ats 36926 df-atl 36957 df-cvlat 36981 df-hlat 37010 df-llines 37157 df-lplanes 37158 df-lvols 37159 df-lines 37160 df-psubsp 37162 df-pmap 37163 df-padd 37455 df-lhyp 37647 df-laut 37648 df-ldil 37763 df-ltrn 37764 df-trl 37818 df-tgrp 38402 df-tendo 38414 df-edring 38416 df-dveca 38662 df-disoa 38688 df-dvech 38738 df-dib 38798 df-dic 38832 df-dih 38888 df-doch 39007 df-djh 39054 |
This theorem is referenced by: lclkrlem2i 39174 |
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