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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh7cN | Structured version Visualization version GIF version |
Description: Part (7) of [Baer] p. 48 line 10 (3 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) |
Ref | Expression |
---|---|
mapdh7cN | ⊢ (𝜑 → (𝐼‘〈𝑣, 𝐺, 𝑢〉) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh7a | . 2 ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑣〉) = 𝐺) | |
2 | mapdh7.q | . . 3 ⊢ 𝑄 = (0g‘𝐶) | |
3 | mapdh7.i | . . 3 ⊢ 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)}))))) | |
4 | mapdh7.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | mapdh7.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
6 | mapdh7.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | mapdh7.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
8 | mapdh7.s | . . 3 ⊢ − = (-g‘𝑈) | |
9 | mapdh7.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
10 | mapdh7.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
11 | mapdh7.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
12 | mapdh7.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
13 | mapdh7.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
14 | mapdh7.j | . . 3 ⊢ 𝐽 = (LSpan‘𝐶) | |
15 | mapdh7.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | mapdh7.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
17 | mapdh7.mn | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹})) | |
18 | mapdh7.x | . . 3 ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) | |
19 | mapdh7.y | . . 3 ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) | |
20 | 19 | eldifad 3959 | . . . . 5 ⊢ (𝜑 → 𝑣 ∈ 𝑉) |
21 | mapdh7.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣})) | |
22 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21 | mapdhcl 41437 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑢, 𝐹, 𝑣〉) ∈ 𝐷) |
23 | 1, 22 | eqeltrrd 2827 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷) |
24 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 23, 21 | mapdheq2 41439 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑢, 𝐹, 𝑣〉) = 𝐺 → (𝐼‘〈𝑣, 𝐺, 𝑢〉) = 𝐹)) |
25 | 1, 24 | mpd 15 | 1 ⊢ (𝜑 → (𝐼‘〈𝑣, 𝐺, 𝑢〉) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3463 ∖ cdif 3944 ifcif 4524 {csn 4624 {cpr 4626 〈cotp 4632 ↦ cmpt 5227 ‘cfv 6544 ℩crio 7369 (class class class)co 7414 1st c1st 7991 2nd c2nd 7992 Basecbs 17206 0gc0g 17447 -gcsg 18923 LSpanclspn 20942 HLchlt 39059 LHypclh 39694 DVecHcdvh 40788 LCDualclcd 41296 mapdcmpd 41334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-riotaBAD 38662 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4907 df-int 4948 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8231 df-undef 8278 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-n0 12517 df-z 12603 df-uz 12867 df-fz 13531 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-mulr 17273 df-sca 17275 df-vsca 17276 df-0g 17449 df-mre 17592 df-mrc 17593 df-acs 17595 df-proset 18313 df-poset 18331 df-plt 18348 df-lub 18364 df-glb 18365 df-join 18366 df-meet 18367 df-p0 18443 df-p1 18444 df-lat 18450 df-clat 18517 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-cntz 19305 df-oppg 19334 df-lsm 19628 df-cmn 19774 df-abl 19775 df-mgp 20112 df-rng 20130 df-ur 20159 df-ring 20212 df-oppr 20310 df-dvdsr 20333 df-unit 20334 df-invr 20364 df-dvr 20377 df-nzr 20489 df-rlreg 20666 df-domn 20667 df-drng 20703 df-lmod 20832 df-lss 20903 df-lsp 20943 df-lvec 21075 df-lsatoms 38685 df-lshyp 38686 df-lcv 38728 df-lfl 38767 df-lkr 38795 df-ldual 38833 df-oposet 38885 df-ol 38887 df-oml 38888 df-covers 38975 df-ats 38976 df-atl 39007 df-cvlat 39031 df-hlat 39060 df-llines 39208 df-lplanes 39209 df-lvols 39210 df-lines 39211 df-psubsp 39213 df-pmap 39214 df-padd 39506 df-lhyp 39698 df-laut 39699 df-ldil 39814 df-ltrn 39815 df-trl 39869 df-tgrp 40453 df-tendo 40465 df-edring 40467 df-dveca 40713 df-disoa 40739 df-dvech 40789 df-dib 40849 df-dic 40883 df-dih 40939 df-doch 41058 df-djh 41105 df-lcdual 41297 df-mapd 41335 |
This theorem is referenced by: (None) |
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