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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh75cN | 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 |
---|---|
mapdh75.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh75.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh75.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh75.s | ⊢ − = (-g‘𝑈) |
mapdh75.o | ⊢ 0 = (0g‘𝑈) |
mapdh75.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh75.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh75.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh75.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh75.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh75.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh75.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh75.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)}))))) |
mapdh75.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdh75.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh75.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdh75a | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
mapdh75c.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdh75c.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh75c.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
mapdh75cN | ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑋〉) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh75.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdh75.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdh75.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
4 | mapdh75.s | . 2 ⊢ − = (-g‘𝑈) | |
5 | mapdh75.o | . 2 ⊢ 0 = (0g‘𝑈) | |
6 | mapdh75.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | mapdh75.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | mapdh75.d | . 2 ⊢ 𝐷 = (Base‘𝐶) | |
9 | mapdh75.r | . 2 ⊢ 𝑅 = (-g‘𝐶) | |
10 | mapdh75.q | . 2 ⊢ 𝑄 = (0g‘𝐶) | |
11 | mapdh75.j | . 2 ⊢ 𝐽 = (LSpan‘𝐶) | |
12 | mapdh75.m | . 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
13 | mapdh75.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 | mapdh75.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | mapdh75.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
16 | mapdh75.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
17 | mapdh75a | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
18 | mapdh75c.ne | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
19 | mapdh75c.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
20 | mapdh75c.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 | mapdh75e 41734 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑋〉) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∖ cdif 3959 ifcif 4530 {csn 4630 〈cotp 4638 ↦ cmpt 5230 ‘cfv 6562 ℩crio 7386 (class class class)co 7430 1st c1st 8010 2nd c2nd 8011 Basecbs 17244 0gc0g 17485 -gcsg 18965 LSpanclspn 20986 HLchlt 39331 LHypclh 39966 DVecHcdvh 41060 LCDualclcd 41568 mapdcmpd 41606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-riotaBAD 38934 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-undef 8296 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-oppg 19376 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-nzr 20529 df-rlreg 20710 df-domn 20711 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-lsatoms 38957 df-lshyp 38958 df-lcv 39000 df-lfl 39039 df-lkr 39067 df-ldual 39105 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 df-tgrp 40725 df-tendo 40737 df-edring 40739 df-dveca 40985 df-disoa 41011 df-dvech 41061 df-dib 41121 df-dic 41155 df-dih 41211 df-doch 41330 df-djh 41377 df-lcdual 41569 df-mapd 41607 |
This theorem is referenced by: (None) |
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