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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap11lem1 | Structured version Visualization version GIF version |
Description: Lemma for hdmapadd 38994. (Contributed by NM, 26-May-2015.) |
Ref | Expression |
---|---|
hdmap11.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap11.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap11.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap11.p | ⊢ + = (+g‘𝑈) |
hdmap11.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap11.a | ⊢ ✚ = (+g‘𝐶) |
hdmap11.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap11.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap11.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmap11.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
hdmap11.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmap11.o | ⊢ 0 = (0g‘𝑈) |
hdmap11.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap11.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap11.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap11.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap11.j | ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) |
hdmap11.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap11lem0.1a | ⊢ (𝜑 → 𝑧 ∈ 𝑉) |
hdmap11lem0.6 | ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
hdmap11lem0.2a | ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸})) |
Ref | Expression |
---|---|
hdmap11lem1 | ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap11.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap11.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap11.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap11.p | . . 3 ⊢ + = (+g‘𝑈) | |
5 | hdmap11.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | hdmap11.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
7 | hdmap11.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap11.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
9 | hdmap11.a | . . 3 ⊢ ✚ = (+g‘𝐶) | |
10 | hdmap11.l | . . 3 ⊢ 𝐿 = (LSpan‘𝐶) | |
11 | hdmap11.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
12 | hdmap11.i | . . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
13 | hdmap11.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
15 | hdmap11.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
16 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2821 | . . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
18 | hdmap11.e | . . . . . . 7 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
19 | 1, 16, 17, 2, 3, 5, 18, 13 | dvheveccl 38263 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 })) |
20 | 1, 2, 3, 5, 7, 8, 14, 15, 13, 19 | hvmapcl2 38917 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
21 | 20 | eldifad 3948 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
22 | 1, 2, 3, 5, 6, 7, 10, 11, 15, 13, 19 | mapdhvmap 38920 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝐸})) = (𝐿‘{(𝐽‘𝐸)})) |
23 | hdmap11lem0.2a | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸})) | |
24 | 23 | necomd 3071 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑧})) |
25 | hdmap11lem0.1a | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝑉) | |
26 | 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 21, 22, 24, 19, 25 | hdmap1cl 38955 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) ∈ 𝐷) |
27 | eqid 2821 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
28 | 1, 2, 13 | dvhlmod 38261 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
29 | hdmap11.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
30 | hdmap11.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
31 | 3, 27, 6, 28, 29, 30 | lspprcl 19750 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
32 | hdmap11lem0.6 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) | |
33 | 5, 27, 28, 31, 25, 32 | lssneln0 19724 | . . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑉 ∖ { 0 })) |
34 | eqidd 2822 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)) | |
35 | eqid 2821 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
36 | eqid 2821 | . . . . . 6 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
37 | 1, 2, 3, 35, 5, 6, 7, 8, 36, 10, 11, 12, 13, 19, 21, 33, 26, 24, 22 | hdmap1eq 38952 | . . . . 5 ⊢ (𝜑 → ((𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) ↔ ((𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)}) ∧ (𝑀‘(𝑁‘{(𝐸(-g‘𝑈)𝑧)})) = (𝐿‘{((𝐽‘𝐸)(-g‘𝐶)(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉))})))) |
38 | 34, 37 | mpbid 234 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)}) ∧ (𝑀‘(𝑁‘{(𝐸(-g‘𝑈)𝑧)})) = (𝐿‘{((𝐽‘𝐸)(-g‘𝐶)(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉))}))) |
39 | 38 | simpld 497 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)})) |
40 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 33, 29, 30, 32, 39 | hdmap1l6 38972 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), (𝑋 + 𝑌)〉) = ((𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉) ✚ (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉))) |
41 | hdmap11.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
42 | 3, 4 | lmodvacl 19648 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
43 | 28, 29, 30, 42 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
44 | 1, 2, 13 | dvhlvec 38260 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec) |
45 | 19 | eldifad 3948 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
46 | 3, 4, 6, 28, 29, 30 | lspprvacl 19771 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌})) |
47 | 27, 6, 28, 31, 46 | lspsnel5a 19768 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌})) |
48 | 47, 32 | ssneldd 3970 | . . . . 5 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{(𝑋 + 𝑌)})) |
49 | 3, 6, 28, 25, 43, 48 | lspsnne2 19890 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{(𝑋 + 𝑌)})) |
50 | 3, 6, 5, 44, 45, 43, 33, 23, 49 | hdmaplem4 38925 | . . 3 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{(𝑋 + 𝑌)}))) |
51 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 43, 25, 50 | hdmapval2 38983 | . 2 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), (𝑋 + 𝑌)〉)) |
52 | 3, 6, 44, 25, 29, 30, 32 | lspindpi 19904 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌}))) |
53 | 52 | simpld 497 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) |
54 | 3, 6, 5, 44, 45, 29, 33, 23, 53 | hdmaplem4 38925 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑋}))) |
55 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 29, 25, 54 | hdmapval2 38983 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉)) |
56 | 52 | simprd 498 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌})) |
57 | 3, 6, 5, 44, 45, 30, 33, 23, 56 | hdmaplem4 38925 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑌}))) |
58 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 30, 25, 57 | hdmapval2 38983 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉)) |
59 | 55, 58 | oveq12d 7174 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋) ✚ (𝑆‘𝑌)) = ((𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉) ✚ (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉))) |
60 | 40, 51, 59 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 {csn 4567 {cpr 4569 〈cop 4573 〈cotp 4575 I cid 5459 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 -gcsg 18105 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 HLchlt 36501 LHypclh 37135 LTrncltrn 37252 DVecHcdvh 38229 LCDualclcd 38737 mapdcmpd 38775 HVMapchvm 38907 HDMap1chdma1 38942 HDMapchdma 38943 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-mre 16857 df-mrc 16858 df-acs 16860 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-oppg 18474 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lsatoms 36127 df-lshyp 36128 df-lcv 36170 df-lfl 36209 df-lkr 36237 df-ldual 36275 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tgrp 37894 df-tendo 37906 df-edring 37908 df-dveca 38154 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 df-doch 38499 df-djh 38546 df-lcdual 38738 df-mapd 38776 df-hvmap 38908 df-hdmap1 38944 df-hdmap 38945 |
This theorem is referenced by: hdmap11lem2 38993 |
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