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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6bN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 37903. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdh.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh.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)}))))) |
mapdh.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh.s | ⊢ − = (-g‘𝑈) |
mapdhc.o | ⊢ 0 = (0g‘𝑈) |
mapdh.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdhc.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdhcl.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh.p | ⊢ + = (+g‘𝑈) |
mapdh.a | ⊢ ✚ = (+g‘𝐶) |
mapdh6b.y | ⊢ (𝜑 → 𝑌 = 0 ) |
mapdh6b.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
mapdh6b.ne | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
Ref | Expression |
---|---|
mapdh6bN | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdh.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
3 | mapdh.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 37748 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | lmodgrp 19262 | . . . 4 ⊢ (𝐶 ∈ LMod → 𝐶 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Grp) |
7 | mapdh.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
8 | mapdh.i | . . . 4 ⊢ 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)}))))) | |
9 | mapdh.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
10 | mapdh.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
11 | mapdh.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
12 | mapdh.s | . . . 4 ⊢ − = (-g‘𝑈) | |
13 | mapdhc.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
14 | mapdh.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
15 | mapdh.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
16 | mapdh.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
17 | mapdh.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
18 | mapdhc.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
19 | mapdh.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
20 | mapdhcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
21 | mapdh6b.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
22 | 1, 10, 3 | dvhlvec 37265 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 20 | eldifad 3804 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | mapdh6b.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 = 0 ) | |
25 | 1, 10, 3 | dvhlmod 37266 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
26 | 11, 13 | lmod0vcl 19284 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉) |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑉) |
28 | 24, 27 | eqeltrd 2859 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
29 | mapdh6b.ne | . . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
30 | 11, 14, 22, 23, 28, 21, 29 | lspindpi 19528 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
31 | 30 | simprd 491 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
32 | 7, 8, 1, 9, 10, 11, 12, 13, 14, 2, 15, 16, 17, 3, 18, 19, 20, 21, 31 | mapdhcl 37883 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
33 | mapdh.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
34 | 15, 33, 7 | grplid 17839 | . . 3 ⊢ ((𝐶 ∈ Grp ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) → (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
35 | 6, 32, 34 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
36 | 24 | oteq3d 4650 | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝐹, 𝑌〉 = 〈𝑋, 𝐹, 0 〉) |
37 | 36 | fveq2d 6450 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐼‘〈𝑋, 𝐹, 0 〉)) |
38 | 7, 8, 13, 20, 18 | mapdhval0 37881 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) |
39 | 37, 38 | eqtrd 2814 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝑄) |
40 | 39 | oveq1d 6937 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
41 | 24 | oveq1d 6937 | . . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) = ( 0 + 𝑍)) |
42 | lmodgrp 19262 | . . . . . . 7 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp) | |
43 | 25, 42 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Grp) |
44 | mapdh.p | . . . . . . 7 ⊢ + = (+g‘𝑈) | |
45 | 11, 44, 13 | grplid 17839 | . . . . . 6 ⊢ ((𝑈 ∈ Grp ∧ 𝑍 ∈ 𝑉) → ( 0 + 𝑍) = 𝑍) |
46 | 43, 21, 45 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ( 0 + 𝑍) = 𝑍) |
47 | 41, 46 | eqtrd 2814 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = 𝑍) |
48 | 47 | oteq3d 4650 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝐹, (𝑌 + 𝑍)〉 = 〈𝑋, 𝐹, 𝑍〉) |
49 | 48 | fveq2d 6450 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
50 | 35, 40, 49 | 3eqtr4rd 2825 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 Vcvv 3398 ∖ cdif 3789 ifcif 4307 {csn 4398 {cpr 4400 〈cotp 4406 ↦ cmpt 4965 ‘cfv 6135 ℩crio 6882 (class class class)co 6922 1st c1st 7443 2nd c2nd 7444 Basecbs 16255 +gcplusg 16338 0gc0g 16486 Grpcgrp 17809 -gcsg 17811 LModclmod 19255 LSpanclspn 19366 HLchlt 35506 LHypclh 36140 DVecHcdvh 37234 LCDualclcd 37742 mapdcmpd 37780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35109 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-mre 16632 df-mrc 16633 df-acs 16635 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-oppg 18159 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35132 df-lshyp 35133 df-lcv 35175 df-lfl 35214 df-lkr 35242 df-ldual 35280 df-oposet 35332 df-ol 35334 df-oml 35335 df-covers 35422 df-ats 35423 df-atl 35454 df-cvlat 35478 df-hlat 35507 df-llines 35654 df-lplanes 35655 df-lvols 35656 df-lines 35657 df-psubsp 35659 df-pmap 35660 df-padd 35952 df-lhyp 36144 df-laut 36145 df-ldil 36260 df-ltrn 36261 df-trl 36315 df-tgrp 36899 df-tendo 36911 df-edring 36913 df-dveca 37159 df-disoa 37185 df-dvech 37235 df-dib 37295 df-dic 37329 df-dih 37385 df-doch 37504 df-djh 37551 df-lcdual 37743 df-mapd 37781 |
This theorem is referenced by: mapdh6kN 37902 |
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