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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem42 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 39293. Eliminate nonzero condition. (Contributed by NM, 11-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem38.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfrlem38.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfrlem38.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfrlem38.p | ⊢ + = (+g‘𝑈) |
lcfrlem38.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfrlem38.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfrlem38.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcfrlem38.q | ⊢ 𝑄 = (LSubSp‘𝐷) |
lcfrlem38.c | ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcfrlem38.e | ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) |
lcfrlem38.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem38.g | ⊢ (𝜑 → 𝐺 ∈ 𝑄) |
lcfrlem38.gs | ⊢ (𝜑 → 𝐺 ⊆ 𝐶) |
lcfrlem38.xe | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
lcfrlem38.ye | ⊢ (𝜑 → 𝑌 ∈ 𝐸) |
Ref | Expression |
---|---|
lcfrlem42 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem38.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcfrlem38.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | lcfrlem38.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlmod 38818 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | lcfrlem38.o | . . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
6 | eqid 2734 | . . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
7 | lcfrlem38.l | . . . . . 6 ⊢ 𝐿 = (LKer‘𝑈) | |
8 | lcfrlem38.d | . . . . . 6 ⊢ 𝐷 = (LDual‘𝑈) | |
9 | lcfrlem38.q | . . . . . 6 ⊢ 𝑄 = (LSubSp‘𝐷) | |
10 | lcfrlem38.e | . . . . . 6 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) | |
11 | lcfrlem38.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑄) | |
12 | lcfrlem38.xe | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
13 | 1, 5, 2, 6, 7, 8, 9, 10, 3, 11, 12 | lcfrlem4 39253 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈)) |
14 | lcfrlem38.ye | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐸) | |
15 | 1, 5, 2, 6, 7, 8, 9, 10, 3, 11, 14 | lcfrlem4 39253 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈)) |
16 | lcfrlem38.p | . . . . . 6 ⊢ + = (+g‘𝑈) | |
17 | 6, 16 | lmodcom 19917 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑈) ∧ 𝑌 ∈ (Base‘𝑈)) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
18 | 4, 13, 15, 17 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
19 | 18 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
20 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
21 | 11 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → 𝐺 ∈ 𝑄) |
22 | 14 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → 𝑌 ∈ 𝐸) |
23 | eqid 2734 | . . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
24 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → 𝑋 = (0g‘𝑈)) | |
25 | 1, 5, 2, 16, 7, 8, 9, 20, 21, 10, 22, 23, 24 | lcfrlem7 39256 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝑌 + 𝑋) ∈ 𝐸) |
26 | 19, 25 | eqeltrd 2834 | . 2 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝑋 + 𝑌) ∈ 𝐸) |
27 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
28 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → 𝐺 ∈ 𝑄) |
29 | 12 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → 𝑋 ∈ 𝐸) |
30 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → 𝑌 = (0g‘𝑈)) | |
31 | 1, 5, 2, 16, 7, 8, 9, 27, 28, 10, 29, 23, 30 | lcfrlem7 39256 | . 2 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → (𝑋 + 𝑌) ∈ 𝐸) |
32 | lcfrlem38.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
33 | lcfrlem38.c | . . 3 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
34 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
35 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝐺 ∈ 𝑄) |
36 | lcfrlem38.gs | . . . 4 ⊢ (𝜑 → 𝐺 ⊆ 𝐶) | |
37 | 36 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝐺 ⊆ 𝐶) |
38 | 12 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑋 ∈ 𝐸) |
39 | 14 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑌 ∈ 𝐸) |
40 | simprl 771 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑋 ≠ (0g‘𝑈)) | |
41 | simprr 773 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑌 ≠ (0g‘𝑈)) | |
42 | 1, 5, 2, 16, 32, 7, 8, 9, 33, 10, 34, 35, 37, 38, 39, 23, 40, 41 | lcfrlem41 39291 | . 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → (𝑋 + 𝑌) ∈ 𝐸) |
43 | 26, 31, 42 | pm2.61da2ne 3023 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 {crab 3058 ⊆ wss 3857 ∪ ciun 4894 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 +gcplusg 16767 0gc0g 16916 LModclmod 19871 LSubSpclss 19940 LFnlclfn 36765 LKerclk 36793 LDualcld 36831 HLchlt 37058 LHypclh 37692 DVecHcdvh 38786 ocHcoch 39055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-riotaBAD 36661 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-tpos 7957 df-undef 8004 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-0g 16918 df-mre 17061 df-mrc 17062 df-acs 17064 df-proset 17774 df-poset 17792 df-plt 17808 df-lub 17824 df-glb 17825 df-join 17826 df-meet 17827 df-p0 17903 df-p1 17904 df-lat 17910 df-clat 17977 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-cntz 18683 df-oppg 18710 df-lsm 18997 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-oppr 19613 df-dvdsr 19631 df-unit 19632 df-invr 19662 df-dvr 19673 df-drng 19741 df-lmod 19873 df-lss 19941 df-lsp 19981 df-lvec 20112 df-lsatoms 36684 df-lshyp 36685 df-lcv 36727 df-lfl 36766 df-lkr 36794 df-ldual 36832 df-oposet 36884 df-ol 36886 df-oml 36887 df-covers 36974 df-ats 36975 df-atl 37006 df-cvlat 37030 df-hlat 37059 df-llines 37206 df-lplanes 37207 df-lvols 37208 df-lines 37209 df-psubsp 37211 df-pmap 37212 df-padd 37504 df-lhyp 37696 df-laut 37697 df-ldil 37812 df-ltrn 37813 df-trl 37867 df-tgrp 38451 df-tendo 38463 df-edring 38465 df-dveca 38711 df-disoa 38737 df-dvech 38787 df-dib 38847 df-dic 38881 df-dih 38937 df-doch 39056 df-djh 39103 |
This theorem is referenced by: lcfr 39293 |
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