![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem1a | Structured version Visualization version GIF version |
Description: Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.) |
Ref | Expression |
---|---|
hdmap14lem1a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem1a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem1a.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem1a.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem1a.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem1a.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem1a.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem2a.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem1a.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap14lem2a.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem2a.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem1a.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem1a.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem3a.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmap14lem1a.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
hdmap14lem1a.z | ⊢ 0 = (0g‘𝑅) |
hdmap14lem1a.fn | ⊢ (𝜑 → 𝐹 ≠ 0 ) |
Ref | Expression |
---|---|
hdmap14lem1a | ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem1a.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap14lem1a.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap14lem1a.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlvec 41091 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec) |
5 | hdmap14lem1a.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
6 | hdmap14lem1a.fn | . . . 4 ⊢ (𝜑 → 𝐹 ≠ 0 ) | |
7 | hdmap14lem3a.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
8 | hdmap14lem1a.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
9 | hdmap14lem1a.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
10 | hdmap14lem1a.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑈) | |
11 | hdmap14lem1a.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
12 | hdmap14lem1a.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
13 | eqid 2734 | . . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
14 | 8, 9, 10, 11, 12, 13 | lspsnvs 21133 | . . . 4 ⊢ ((𝑈 ∈ LVec ∧ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑈)‘{(𝐹 · 𝑋)}) = ((LSpan‘𝑈)‘{𝑋})) |
15 | 4, 5, 6, 7, 14 | syl121anc 1374 | . . 3 ⊢ (𝜑 → ((LSpan‘𝑈)‘{(𝐹 · 𝑋)}) = ((LSpan‘𝑈)‘{𝑋})) |
16 | 15 | fveq2d 6910 | . 2 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{(𝐹 · 𝑋)})) = (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{𝑋}))) |
17 | hdmap14lem1a.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
18 | hdmap14lem1a.l | . . 3 ⊢ 𝐿 = (LSpan‘𝐶) | |
19 | eqid 2734 | . . 3 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊) | |
20 | hdmap14lem1a.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
21 | 1, 2, 3 | dvhlmod 41092 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
22 | 8, 9, 10, 11 | lmodvscl 20892 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉) |
23 | 21, 5, 7, 22 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉) |
24 | 1, 2, 8, 13, 17, 18, 19, 20, 3, 23 | hdmap10 41822 | . 2 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{(𝐹 · 𝑋)})) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))})) |
25 | 1, 2, 8, 13, 17, 18, 19, 20, 3, 7 | hdmap10 41822 | . 2 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{𝑋})) = (𝐿‘{(𝑆‘𝑋)})) |
26 | 16, 24, 25 | 3eqtr3rd 2783 | 1 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 {csn 4630 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17485 LModclmod 20874 LSpanclspn 20986 LVecclvec 21118 HLchlt 39331 LHypclh 39966 DVecHcdvh 41060 LCDualclcd 41568 mapdcmpd 41606 HDMapchdma 41774 |
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 df-hvmap 41739 df-hdmap1 41775 df-hdmap 41776 |
This theorem is referenced by: hdmap14lem2a 41849 hdmap14lem1 41850 |
Copyright terms: Public domain | W3C validator |