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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh7dN | Structured version Visualization version GIF version |
Description: Part (7) of [Baer] p. 48 line 10 (4 of 6 cases). (Contributed by NM, 2-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdh7.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh7.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh7.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh7.s | ⊢ − = (-g‘𝑈) |
mapdh7.o | ⊢ 0 = (0g‘𝑈) |
mapdh7.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh7.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh7.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh7.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh7.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh7.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh7.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh7.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)}))))) |
mapdh7.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdh7.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh7.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹})) |
mapdh7.x | ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) |
mapdh7.y | ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) |
mapdh7.z | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdh7.ne | ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣})) |
mapdh7.wn | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣})) |
mapdh7a | ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑣〉) = 𝐺) |
mapdh7.b | ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑤〉) = 𝐸) |
Ref | Expression |
---|---|
mapdh7dN | ⊢ (𝜑 → (𝐼‘〈𝑣, 𝐺, 𝑤〉) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh7.q | . 2 ⊢ 𝑄 = (0g‘𝐶) | |
2 | mapdh7.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)}))))) | |
3 | mapdh7.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | mapdh7.m | . 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
5 | mapdh7.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | mapdh7.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
7 | mapdh7.s | . 2 ⊢ − = (-g‘𝑈) | |
8 | mapdh7.o | . 2 ⊢ 0 = (0g‘𝑈) | |
9 | mapdh7.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
10 | mapdh7.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
11 | mapdh7.d | . 2 ⊢ 𝐷 = (Base‘𝐶) | |
12 | mapdh7.r | . 2 ⊢ 𝑅 = (-g‘𝐶) | |
13 | mapdh7.j | . 2 ⊢ 𝐽 = (LSpan‘𝐶) | |
14 | mapdh7.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | mapdh7.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
16 | mapdh7.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹})) | |
17 | mapdh7.x | . 2 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) | |
18 | mapdh7.y | . 2 ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) | |
19 | mapdh7.z | . 2 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
20 | 3, 5, 14 | dvhlvec 38247 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec) |
21 | 18 | eldifad 3950 | . . . . 5 ⊢ (𝜑 → 𝑣 ∈ 𝑉) |
22 | 19 | eldifad 3950 | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
23 | mapdh7.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣})) | |
24 | mapdh7.wn | . . . . 5 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣})) | |
25 | 6, 8, 9, 20, 17, 21, 22, 23, 24 | lspindp1 19907 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑣}) ∧ ¬ 𝑢 ∈ (𝑁‘{𝑤, 𝑣}))) |
26 | 25 | simprd 498 | . . 3 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑤, 𝑣})) |
27 | prcom 4670 | . . . . 5 ⊢ {𝑣, 𝑤} = {𝑤, 𝑣} | |
28 | 27 | fveq2i 6675 | . . . 4 ⊢ (𝑁‘{𝑣, 𝑤}) = (𝑁‘{𝑤, 𝑣}) |
29 | 28 | eleq2i 2906 | . . 3 ⊢ (𝑢 ∈ (𝑁‘{𝑣, 𝑤}) ↔ 𝑢 ∈ (𝑁‘{𝑤, 𝑣})) |
30 | 26, 29 | sylnibr 331 | . 2 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣, 𝑤})) |
31 | 17 | eldifad 3950 | . . . . 5 ⊢ (𝜑 → 𝑢 ∈ 𝑉) |
32 | 6, 9, 20, 22, 31, 21, 24 | lspindpi 19906 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑢}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣}))) |
33 | 32 | simprd 498 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑣})) |
34 | 33 | necomd 3073 | . 2 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (𝑁‘{𝑤})) |
35 | mapdh7a | . 2 ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑣〉) = 𝐺) | |
36 | mapdh7.b | . 2 ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑤〉) = 𝐸) | |
37 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 34, 35, 36 | mapdheq4 38870 | 1 ⊢ (𝜑 → (𝐼‘〈𝑣, 𝐺, 𝑤〉) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∖ cdif 3935 ifcif 4469 {csn 4569 {cpr 4571 〈cotp 4577 ↦ cmpt 5148 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 Basecbs 16485 0gc0g 16715 -gcsg 18107 LSpanclspn 19745 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 LCDualclcd 38724 mapdcmpd 38762 |
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 |
This theorem is referenced by: mapdh7fN 38889 |
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