Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem3N | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 15, T ≠ P. Our (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmaprnlem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmaprnlem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmaprnlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmaprnlem1.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmaprnlem1.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmaprnlem1.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmaprnlem1.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmaprnlem1.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmaprnlem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmaprnlem1.se | ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) |
hdmaprnlem1.ve | ⊢ (𝜑 → 𝑣 ∈ 𝑉) |
hdmaprnlem1.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) |
hdmaprnlem1.ue | ⊢ (𝜑 → 𝑢 ∈ 𝑉) |
hdmaprnlem1.un | ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) |
hdmaprnlem1.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmaprnlem1.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmaprnlem1.o | ⊢ 0 = (0g‘𝑈) |
hdmaprnlem1.a | ⊢ ✚ = (+g‘𝐶) |
Ref | Expression |
---|---|
hdmaprnlem3N | ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmaprnlem1.d | . . . . 5 ⊢ 𝐷 = (Base‘𝐶) | |
2 | hdmaprnlem1.l | . . . . 5 ⊢ 𝐿 = (LSpan‘𝐶) | |
3 | hdmaprnlem1.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | hdmaprnlem1.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
5 | hdmaprnlem1.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 3, 4, 5 | lcdlmod 38730 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod) |
7 | hdmaprnlem1.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | hdmaprnlem1.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
9 | hdmaprnlem1.s | . . . . . . 7 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
10 | hdmaprnlem1.ue | . . . . . . 7 ⊢ (𝜑 → 𝑢 ∈ 𝑉) | |
11 | 3, 7, 8, 4, 1, 9, 5, 10 | hdmapcl 38968 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷) |
12 | hdmaprnlem1.se | . . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) | |
13 | 12 | eldifad 3950 | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ 𝐷) |
14 | hdmaprnlem1.a | . . . . . . 7 ⊢ ✚ = (+g‘𝐶) | |
15 | 1, 14 | lmodvacl 19650 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
16 | 6, 11, 13, 15 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) |
17 | eqid 2823 | . . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
18 | 1, 17, 2 | lspsncl 19751 | . . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
19 | 6, 13, 18 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐿‘{𝑠}) ∈ (LSubSp‘𝐶)) |
20 | 1, 2 | lspsnid 19767 | . . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝐿‘{𝑠})) |
21 | 6, 13, 20 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝑠 ∈ (𝐿‘{𝑠})) |
22 | hdmaprnlem1.q | . . . . . . 7 ⊢ 𝑄 = (0g‘𝐶) | |
23 | 3, 4, 5 | lcdlvec 38729 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ LVec) |
24 | hdmaprnlem1.o | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
25 | eqid 2823 | . . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
26 | 3, 7, 5 | dvhlmod 38248 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
27 | hdmaprnlem1.ve | . . . . . . . . . 10 ⊢ (𝜑 → 𝑣 ∈ 𝑉) | |
28 | hdmaprnlem1.n | . . . . . . . . . . 11 ⊢ 𝑁 = (LSpan‘𝑈) | |
29 | 8, 25, 28 | lspsncl 19751 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
30 | 26, 27, 29 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) |
31 | hdmaprnlem1.un | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) | |
32 | 24, 25, 26, 30, 10, 31 | lssneln0 19726 | . . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) |
33 | 3, 7, 8, 24, 4, 22, 1, 9, 5, 32 | hdmapnzcl 38983 | . . . . . . 7 ⊢ (𝜑 → (𝑆‘𝑢) ∈ (𝐷 ∖ {𝑄})) |
34 | hdmaprnlem1.m | . . . . . . . 8 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
35 | hdmaprnlem1.e | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) | |
36 | 3, 7, 8, 28, 4, 2, 34, 9, 5, 12, 27, 35, 10, 31 | hdmaprnlem1N 38987 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠})) |
37 | 1, 22, 2, 23, 33, 13, 36 | lspsnne1 19891 | . . . . . 6 ⊢ (𝜑 → ¬ (𝑆‘𝑢) ∈ (𝐿‘{𝑠})) |
38 | 1, 14, 17, 6, 19, 21, 11, 37 | lssvancl2 19719 | . . . . 5 ⊢ (𝜑 → ¬ ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐿‘{𝑠})) |
39 | 1, 2, 6, 16, 13, 38 | lspsnne2 19892 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ≠ (𝐿‘{𝑠})) |
40 | 39 | necomd 3073 | . . 3 ⊢ (𝜑 → (𝐿‘{𝑠}) ≠ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) |
41 | 1, 17, 2 | lspsncl 19751 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
42 | 6, 16, 41 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶)) |
43 | 3, 34, 4, 17, 5 | mapdrn2 38789 | . . . . 5 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶)) |
44 | 42, 43 | eleqtrrd 2918 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀) |
45 | 3, 34, 5, 44 | mapdcnvid2 38795 | . . 3 ⊢ (𝜑 → (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) = (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) |
46 | 40, 35, 45 | 3netr4d 3095 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) |
47 | 3, 34, 7, 25, 5, 44 | mapdcnvcl 38790 | . . . 4 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSubSp‘𝑈)) |
48 | 3, 7, 25, 34, 5, 30, 47 | mapd11 38777 | . . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) = (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) = (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) |
49 | 48 | necon3bid 3062 | . 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑣})) ≠ (𝑀‘(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) ↔ (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))) |
50 | 46, 49 | mpbid 234 | 1 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 ◡ccnv 5556 ran crn 5558 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 0gc0g 16715 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 LCDualclcd 38724 mapdcmpd 38762 HDMapchdma 38930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-lcv 36157 df-lfl 36196 df-lkr 36224 df-ldual 36262 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 df-lcdual 38725 df-mapd 38763 df-hvmap 38895 df-hdmap1 38931 df-hdmap 38932 |
This theorem is referenced by: hdmaprnlem9N 38995 hdmaprnlem3eN 38996 |
Copyright terms: Public domain | W3C validator |