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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap11lem1 | Structured version Visualization version GIF version |
Description: Lemma for hdmapadd 39883. (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 2733 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
15 | hdmap11.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
16 | eqid 2733 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2733 | . . . . . . 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 39152 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 })) |
20 | 1, 2, 3, 5, 7, 8, 14, 15, 13, 19 | hvmapcl2 39806 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
21 | 20 | eldifad 3901 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
22 | 1, 2, 3, 5, 6, 7, 10, 11, 15, 13, 19 | mapdhvmap 39809 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝐸})) = (𝐿‘{(𝐽‘𝐸)})) |
23 | hdmap11lem0.2a | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸})) | |
24 | 23 | necomd 2994 | . . . 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 39844 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) ∈ 𝐷) |
27 | eqid 2733 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
28 | 1, 2, 13 | dvhlmod 39150 | . . . 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 20268 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
32 | hdmap11lem0.6 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) | |
33 | 5, 27, 28, 31, 25, 32 | lssneln0 20242 | . . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑉 ∖ { 0 })) |
34 | eqidd 2734 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)) | |
35 | eqid 2733 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
36 | eqid 2733 | . . . . . 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 39841 | . . . . 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 231 | . . . 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 494 | . . 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 39861 | . 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 20165 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
43 | 28, 29, 30, 42 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
44 | 1, 2, 13 | dvhlvec 39149 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec) |
45 | 19 | eldifad 3901 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
46 | 3, 4, 6, 28, 29, 30 | lspprvacl 20289 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌})) |
47 | 27, 6, 28, 31, 46 | lspsnel5a 20286 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌})) |
48 | 47, 32 | ssneldd 3926 | . . . . 5 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{(𝑋 + 𝑌)})) |
49 | 3, 6, 28, 25, 43, 48 | lspsnne2 20408 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{(𝑋 + 𝑌)})) |
50 | 3, 6, 5, 44, 45, 43, 33, 23, 49 | hdmaplem4 39814 | . . 3 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{(𝑋 + 𝑌)}))) |
51 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 43, 25, 50 | hdmapval2 39872 | . 2 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), (𝑋 + 𝑌)〉)) |
52 | 3, 6, 44, 25, 29, 30, 32 | lspindpi 20422 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌}))) |
53 | 52 | simpld 494 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) |
54 | 3, 6, 5, 44, 45, 29, 33, 23, 53 | hdmaplem4 39814 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑋}))) |
55 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 29, 25, 54 | hdmapval2 39872 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉)) |
56 | 52 | simprd 495 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌})) |
57 | 3, 6, 5, 44, 45, 30, 33, 23, 56 | hdmaplem4 39814 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑌}))) |
58 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 30, 25, 57 | hdmapval2 39872 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉)) |
59 | 55, 58 | oveq12d 7313 | . 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 2783 | 1 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ≠ wne 2938 {csn 4564 {cpr 4566 〈cop 4570 〈cotp 4572 I cid 5490 ↾ cres 5593 ‘cfv 6447 (class class class)co 7295 Basecbs 16940 +gcplusg 16990 0gc0g 17178 -gcsg 18607 LModclmod 20151 LSubSpclss 20221 LSpanclspn 20261 HLchlt 37390 LHypclh 38024 LTrncltrn 38141 DVecHcdvh 39118 LCDualclcd 39626 mapdcmpd 39664 HVMapchvm 39796 HDMap1chdma1 39831 HDMapchdma 39832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-riotaBAD 36993 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-ot 4573 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-om 7733 df-1st 7851 df-2nd 7852 df-tpos 8062 df-undef 8109 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-n0 12262 df-z 12348 df-uz 12611 df-fz 13268 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-sca 17006 df-vsca 17007 df-0g 17180 df-mre 17323 df-mrc 17324 df-acs 17326 df-proset 18041 df-poset 18059 df-plt 18076 df-lub 18092 df-glb 18093 df-join 18094 df-meet 18095 df-p0 18171 df-p1 18172 df-lat 18178 df-clat 18245 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-submnd 18459 df-grp 18608 df-minusg 18609 df-sbg 18610 df-subg 18780 df-cntz 18951 df-oppg 18978 df-lsm 19269 df-cmn 19416 df-abl 19417 df-mgp 19749 df-ur 19766 df-ring 19813 df-oppr 19890 df-dvdsr 19911 df-unit 19912 df-invr 19942 df-dvr 19953 df-drng 20021 df-lmod 20153 df-lss 20222 df-lsp 20262 df-lvec 20393 df-lsatoms 37016 df-lshyp 37017 df-lcv 37059 df-lfl 37098 df-lkr 37126 df-ldual 37164 df-oposet 37216 df-ol 37218 df-oml 37219 df-covers 37306 df-ats 37307 df-atl 37338 df-cvlat 37362 df-hlat 37391 df-llines 37538 df-lplanes 37539 df-lvols 37540 df-lines 37541 df-psubsp 37543 df-pmap 37544 df-padd 37836 df-lhyp 38028 df-laut 38029 df-ldil 38144 df-ltrn 38145 df-trl 38199 df-tgrp 38783 df-tendo 38795 df-edring 38797 df-dveca 39043 df-disoa 39069 df-dvech 39119 df-dib 39179 df-dic 39213 df-dih 39269 df-doch 39388 df-djh 39435 df-lcdual 39627 df-mapd 39665 df-hvmap 39797 df-hdmap1 39833 df-hdmap 39834 |
This theorem is referenced by: hdmap11lem2 39882 |
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