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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh8ab | Structured version Visualization version GIF version |
Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-May-2015.) |
Ref | Expression |
---|---|
mapdh8a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh8a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh8a.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh8a.s | ⊢ − = (-g‘𝑈) |
mapdh8a.o | ⊢ 0 = (0g‘𝑈) |
mapdh8a.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh8a.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh8a.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh8a.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh8a.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh8a.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh8a.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh8a.i | ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) |
mapdh8a.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdh8ab.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh8ab.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdh8ab.eg | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
mapdh8ab.ee | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) |
mapdh8ab.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh8ab.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh8ab.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdh8ab.t | ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) |
mapdh8ab.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
mapdh8ab.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
mapdh8ab.yn | ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) |
Ref | Expression |
---|---|
mapdh8ab | ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh8a.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdh8a.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdh8a.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
4 | mapdh8a.s | . 2 ⊢ − = (-g‘𝑈) | |
5 | mapdh8a.o | . 2 ⊢ 0 = (0g‘𝑈) | |
6 | mapdh8a.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | mapdh8a.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | mapdh8a.d | . 2 ⊢ 𝐷 = (Base‘𝐶) | |
9 | mapdh8a.r | . 2 ⊢ 𝑅 = (-g‘𝐶) | |
10 | mapdh8a.q | . 2 ⊢ 𝑄 = (0g‘𝐶) | |
11 | mapdh8a.j | . 2 ⊢ 𝐽 = (LSpan‘𝐶) | |
12 | mapdh8a.m | . 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | mapdh8a.i | . 2 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) | |
14 | mapdh8a.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | mapdh8ab.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
16 | mapdh8ab.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
17 | mapdh8ab.eg | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
18 | mapdh8ab.ee | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) | |
19 | mapdh8ab.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
20 | mapdh8ab.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
21 | mapdh8ab.z | . 2 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
22 | 1, 2, 14 | dvhlvec 38746 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 19 | eldifad 3855 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | 20 | eldifad 3855 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
25 | 21 | eldifad 3855 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
26 | mapdh8ab.xn | . . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
27 | 3, 6, 22, 23, 24, 25, 26 | lspindpi 20023 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
28 | 27 | simprd 499 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
29 | 28 | necomd 2989 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋})) |
30 | mapdh8ab.yn | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) | |
31 | 29, 30 | neeqtrd 3003 | . 2 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) |
32 | mapdh8ab.t | . 2 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) | |
33 | 30 | sseq1d 3908 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
34 | eqid 2738 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
35 | 1, 2, 14 | dvhlmod 38747 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
36 | 3, 34, 6, 35, 24, 25 | lspprcl 19869 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈)) |
37 | 3, 34, 6, 35, 36, 23 | lspsnel5 19886 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
38 | 32 | eldifad 3855 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
39 | 3, 34, 6, 35, 36, 38 | lspsnel5 19886 | . . . . 5 ⊢ (𝜑 → (𝑇 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑍}))) |
40 | 33, 37, 39 | 3bitr4d 314 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ 𝑇 ∈ (𝑁‘{𝑌, 𝑍}))) |
41 | 26, 40 | mtbid 327 | . . 3 ⊢ (𝜑 → ¬ 𝑇 ∈ (𝑁‘{𝑌, 𝑍})) |
42 | 22 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑈 ∈ LVec) |
43 | 20 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
44 | 38 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ 𝑉) |
45 | 25 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑍 ∈ 𝑉) |
46 | mapdh8ab.yz | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) | |
47 | 46 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
48 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) | |
49 | prcom 4623 | . . . . . 6 ⊢ {𝑍, 𝑇} = {𝑇, 𝑍} | |
50 | 49 | fveq2i 6677 | . . . . 5 ⊢ (𝑁‘{𝑍, 𝑇}) = (𝑁‘{𝑇, 𝑍}) |
51 | 48, 50 | eleqtrdi 2843 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑇, 𝑍})) |
52 | 3, 5, 6, 42, 43, 44, 45, 47, 51 | lspexch 20020 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ (𝑁‘{𝑌, 𝑍})) |
53 | 41, 52 | mtand 816 | . 2 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) |
54 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 31, 32, 53, 26 | mapdh8aa 39413 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑇〉) = (𝐼‘〈𝑍, 𝐸, 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 Vcvv 3398 ∖ cdif 3840 ⊆ wss 3843 ifcif 4414 {csn 4516 {cpr 4518 〈cotp 4524 ↦ cmpt 5110 ‘cfv 6339 ℩crio 7126 (class class class)co 7170 1st c1st 7712 2nd c2nd 7713 Basecbs 16586 0gc0g 16816 -gcsg 18221 LSubSpclss 19822 LSpanclspn 19862 LVecclvec 19993 HLchlt 36987 LHypclh 37621 DVecHcdvh 38715 LCDualclcd 39223 mapdcmpd 39261 |
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 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-riotaBAD 36590 |
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-ot 4525 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 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-undef 7968 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-0g 16818 df-mre 16960 df-mrc 16961 df-acs 16963 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-p1 17766 df-lat 17772 df-clat 17834 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-cntz 18565 df-oppg 18592 df-lsm 18879 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-drng 19623 df-lmod 19755 df-lss 19823 df-lsp 19863 df-lvec 19994 df-lsatoms 36613 df-lshyp 36614 df-lcv 36656 df-lfl 36695 df-lkr 36723 df-ldual 36761 df-oposet 36813 df-ol 36815 df-oml 36816 df-covers 36903 df-ats 36904 df-atl 36935 df-cvlat 36959 df-hlat 36988 df-llines 37135 df-lplanes 37136 df-lvols 37137 df-lines 37138 df-psubsp 37140 df-pmap 37141 df-padd 37433 df-lhyp 37625 df-laut 37626 df-ldil 37741 df-ltrn 37742 df-trl 37796 df-tgrp 38380 df-tendo 38392 df-edring 38394 df-dveca 38640 df-disoa 38666 df-dvech 38716 df-dib 38776 df-dic 38810 df-dih 38866 df-doch 38985 df-djh 39032 df-lcdual 39224 df-mapd 39262 |
This theorem is referenced by: mapdh8ac 39415 |
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