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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6bN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 39416. (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 39261 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | lmodgrp 19772 | . . . 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 38778 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 20 | eldifad 3865 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | mapdh6b.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 = 0 ) | |
25 | 1, 10, 3 | dvhlmod 38779 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
26 | 11, 13 | lmod0vcl 19794 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉) |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑉) |
28 | 24, 27 | eqeltrd 2834 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
29 | mapdh6b.ne | . . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
30 | 11, 14, 22, 23, 28, 21, 29 | lspindpi 20035 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
31 | 30 | simprd 499 | . . . 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 39396 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
33 | mapdh.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
34 | 15, 33, 7 | grplid 18263 | . . 3 ⊢ ((𝐶 ∈ Grp ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) → (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
35 | 6, 32, 34 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
36 | 24 | oteq3d 4785 | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝐹, 𝑌〉 = 〈𝑋, 𝐹, 0 〉) |
37 | 36 | fveq2d 6690 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐼‘〈𝑋, 𝐹, 0 〉)) |
38 | 7, 8, 13, 20, 18 | mapdhval0 39394 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) |
39 | 37, 38 | eqtrd 2774 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝑄) |
40 | 39 | oveq1d 7197 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
41 | 24 | oveq1d 7197 | . . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) = ( 0 + 𝑍)) |
42 | lmodgrp 19772 | . . . . . . 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 18263 | . . . . . 6 ⊢ ((𝑈 ∈ Grp ∧ 𝑍 ∈ 𝑉) → ( 0 + 𝑍) = 𝑍) |
46 | 43, 21, 45 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ( 0 + 𝑍) = 𝑍) |
47 | 41, 46 | eqtrd 2774 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = 𝑍) |
48 | 47 | oteq3d 4785 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝐹, (𝑌 + 𝑍)〉 = 〈𝑋, 𝐹, 𝑍〉) |
49 | 48 | fveq2d 6690 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
50 | 35, 40, 49 | 3eqtr4rd 2785 | 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 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 Vcvv 3400 ∖ cdif 3850 ifcif 4424 {csn 4526 {cpr 4528 〈cotp 4534 ↦ cmpt 5120 ‘cfv 6349 ℩crio 7138 (class class class)co 7182 1st c1st 7724 2nd c2nd 7725 Basecbs 16598 +gcplusg 16680 0gc0g 16828 Grpcgrp 18231 -gcsg 18233 LModclmod 19765 LSpanclspn 19874 HLchlt 37019 LHypclh 37653 DVecHcdvh 38747 LCDualclcd 39255 mapdcmpd 39293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-riotaBAD 36622 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-ot 4535 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-om 7612 df-1st 7726 df-2nd 7727 df-tpos 7933 df-undef 7980 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-n0 11989 df-z 12075 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-sca 16696 df-vsca 16697 df-0g 16830 df-mre 16972 df-mrc 16973 df-acs 16975 df-proset 17666 df-poset 17684 df-plt 17696 df-lub 17712 df-glb 17713 df-join 17714 df-meet 17715 df-p0 17777 df-p1 17778 df-lat 17784 df-clat 17846 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-submnd 18085 df-grp 18234 df-minusg 18235 df-sbg 18236 df-subg 18406 df-cntz 18577 df-oppg 18604 df-lsm 18891 df-cmn 19038 df-abl 19039 df-mgp 19371 df-ur 19383 df-ring 19430 df-oppr 19507 df-dvdsr 19525 df-unit 19526 df-invr 19556 df-dvr 19567 df-drng 19635 df-lmod 19767 df-lss 19835 df-lsp 19875 df-lvec 20006 df-lsatoms 36645 df-lshyp 36646 df-lcv 36688 df-lfl 36727 df-lkr 36755 df-ldual 36793 df-oposet 36845 df-ol 36847 df-oml 36848 df-covers 36935 df-ats 36936 df-atl 36967 df-cvlat 36991 df-hlat 37020 df-llines 37167 df-lplanes 37168 df-lvols 37169 df-lines 37170 df-psubsp 37172 df-pmap 37173 df-padd 37465 df-lhyp 37657 df-laut 37658 df-ldil 37773 df-ltrn 37774 df-trl 37828 df-tgrp 38412 df-tendo 38424 df-edring 38426 df-dveca 38672 df-disoa 38698 df-dvech 38748 df-dib 38808 df-dic 38842 df-dih 38898 df-doch 39017 df-djh 39064 df-lcdual 39256 df-mapd 39294 |
This theorem is referenced by: mapdh6kN 39415 |
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