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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6bN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 40934. (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 40779 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | lmodgrp 20625 | . . . 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 40296 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 20 | eldifad 3960 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | mapdh6b.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 = 0 ) | |
25 | 1, 10, 3 | dvhlmod 40297 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
26 | 11, 13 | lmod0vcl 20649 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉) |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑉) |
28 | 24, 27 | eqeltrd 2832 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
29 | mapdh6b.ne | . . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
30 | 11, 14, 22, 23, 28, 21, 29 | lspindpi 20894 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
31 | 30 | simprd 495 | . . . 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 40914 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
33 | mapdh.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
34 | 15, 33, 7 | grplid 18892 | . . 3 ⊢ ((𝐶 ∈ Grp ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) → (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
35 | 6, 32, 34 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
36 | 24 | oteq3d 4887 | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝐹, 𝑌〉 = 〈𝑋, 𝐹, 0 〉) |
37 | 36 | fveq2d 6895 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐼‘〈𝑋, 𝐹, 0 〉)) |
38 | 7, 8, 13, 20, 18 | mapdhval0 40912 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) |
39 | 37, 38 | eqtrd 2771 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝑄) |
40 | 39 | oveq1d 7427 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
41 | 24 | oveq1d 7427 | . . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) = ( 0 + 𝑍)) |
42 | lmodgrp 20625 | . . . . . . 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 18892 | . . . . . 6 ⊢ ((𝑈 ∈ Grp ∧ 𝑍 ∈ 𝑉) → ( 0 + 𝑍) = 𝑍) |
46 | 43, 21, 45 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → ( 0 + 𝑍) = 𝑍) |
47 | 41, 46 | eqtrd 2771 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = 𝑍) |
48 | 47 | oteq3d 4887 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝐹, (𝑌 + 𝑍)〉 = 〈𝑋, 𝐹, 𝑍〉) |
49 | 48 | fveq2d 6895 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
50 | 35, 40, 49 | 3eqtr4rd 2782 | 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 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 ∖ cdif 3945 ifcif 4528 {csn 4628 {cpr 4630 〈cotp 4636 ↦ cmpt 5231 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 1st c1st 7977 2nd c2nd 7978 Basecbs 17151 +gcplusg 17204 0gc0g 17392 Grpcgrp 18858 -gcsg 18860 LModclmod 20618 LSpanclspn 20730 HLchlt 38536 LHypclh 39171 DVecHcdvh 40265 LCDualclcd 40773 mapdcmpd 40811 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38139 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-proset 18255 df-poset 18273 df-plt 18290 df-lub 18306 df-glb 18307 df-join 18308 df-meet 18309 df-p0 18385 df-p1 18386 df-lat 18392 df-clat 18459 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-subg 19043 df-cntz 19226 df-oppg 19255 df-lsm 19549 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-oppr 20229 df-dvdsr 20252 df-unit 20253 df-invr 20283 df-dvr 20296 df-drng 20506 df-lmod 20620 df-lss 20691 df-lsp 20731 df-lvec 20862 df-lsatoms 38162 df-lshyp 38163 df-lcv 38205 df-lfl 38244 df-lkr 38272 df-ldual 38310 df-oposet 38362 df-ol 38364 df-oml 38365 df-covers 38452 df-ats 38453 df-atl 38484 df-cvlat 38508 df-hlat 38537 df-llines 38685 df-lplanes 38686 df-lvols 38687 df-lines 38688 df-psubsp 38690 df-pmap 38691 df-padd 38983 df-lhyp 39175 df-laut 39176 df-ldil 39291 df-ltrn 39292 df-trl 39346 df-tgrp 39930 df-tendo 39942 df-edring 39944 df-dveca 40190 df-disoa 40216 df-dvech 40266 df-dib 40326 df-dic 40360 df-dih 40416 df-doch 40535 df-djh 40582 df-lcdual 40774 df-mapd 40812 |
This theorem is referenced by: mapdh6kN 40933 |
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