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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6bN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 38885. (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 38730 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | lmodgrp 19643 | . . . 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 38247 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 20 | eldifad 3950 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | mapdh6b.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 = 0 ) | |
25 | 1, 10, 3 | dvhlmod 38248 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
26 | 11, 13 | lmod0vcl 19665 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉) |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑉) |
28 | 24, 27 | eqeltrd 2915 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
29 | mapdh6b.ne | . . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
30 | 11, 14, 22, 23, 28, 21, 29 | lspindpi 19906 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
31 | 30 | simprd 498 | . . . 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 38865 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
33 | mapdh.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
34 | 15, 33, 7 | grplid 18135 | . . 3 ⊢ ((𝐶 ∈ Grp ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) → (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
35 | 6, 32, 34 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
36 | 24 | oteq3d 4819 | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝐹, 𝑌〉 = 〈𝑋, 𝐹, 0 〉) |
37 | 36 | fveq2d 6676 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐼‘〈𝑋, 𝐹, 0 〉)) |
38 | 7, 8, 13, 20, 18 | mapdhval0 38863 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) |
39 | 37, 38 | eqtrd 2858 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝑄) |
40 | 39 | oveq1d 7173 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = (𝑄 ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
41 | 24 | oveq1d 7173 | . . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) = ( 0 + 𝑍)) |
42 | lmodgrp 19643 | . . . . . . 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 18135 | . . . . . 6 ⊢ ((𝑈 ∈ Grp ∧ 𝑍 ∈ 𝑉) → ( 0 + 𝑍) = 𝑍) |
46 | 43, 21, 45 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ( 0 + 𝑍) = 𝑍) |
47 | 41, 46 | eqtrd 2858 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = 𝑍) |
48 | 47 | oteq3d 4819 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝐹, (𝑌 + 𝑍)〉 = 〈𝑋, 𝐹, 𝑍〉) |
49 | 48 | fveq2d 6676 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) |
50 | 35, 40, 49 | 3eqtr4rd 2869 | 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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∖ cdif 3935 ifcif 4469 {csn 4569 {cpr 4571 〈cotp 4577 ↦ cmpt 5148 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 Basecbs 16485 +gcplusg 16567 0gc0g 16715 Grpcgrp 18105 -gcsg 18107 LModclmod 19636 LSpanclspn 19745 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 LCDualclcd 38724 mapdcmpd 38762 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-lcv 36157 df-lfl 36196 df-lkr 36224 df-ldual 36262 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 df-lcdual 38725 df-mapd 38763 |
This theorem is referenced by: mapdh6kN 38884 |
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