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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap11lem1 | Structured version Visualization version GIF version |
Description: Lemma for hdmapadd 41826. (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 2735 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
15 | hdmap11.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
16 | eqid 2735 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2735 | . . . . . . 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 41095 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 })) |
20 | 1, 2, 3, 5, 7, 8, 14, 15, 13, 19 | hvmapcl2 41749 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
21 | 20 | eldifad 3975 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
22 | 1, 2, 3, 5, 6, 7, 10, 11, 15, 13, 19 | mapdhvmap 41752 | . . . 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 41787 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) ∈ 𝐷) |
27 | eqid 2735 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
28 | 1, 2, 13 | dvhlmod 41093 | . . . 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 20994 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
32 | hdmap11lem0.6 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) | |
33 | 5, 27, 28, 31, 25, 32 | lssneln0 20969 | . . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑉 ∖ { 0 })) |
34 | eqidd 2736 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)) | |
35 | eqid 2735 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
36 | eqid 2735 | . . . . . 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 41784 | . . . . 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 232 | . . . 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 41804 | . 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 20890 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
43 | 28, 29, 30, 42 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
44 | 1, 2, 13 | dvhlvec 41092 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec) |
45 | 19 | eldifad 3975 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
46 | 3, 4, 6, 28, 29, 30 | lspprvacl 21015 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌})) |
47 | 27, 6, 28, 31, 46 | ellspsn5 21012 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌})) |
48 | 47, 32 | ssneldd 3998 | . . . . 5 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{(𝑋 + 𝑌)})) |
49 | 3, 6, 28, 25, 43, 48 | lspsnne2 21138 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{(𝑋 + 𝑌)})) |
50 | 3, 6, 5, 44, 45, 43, 33, 23, 49 | hdmaplem4 41757 | . . 3 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{(𝑋 + 𝑌)}))) |
51 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 43, 25, 50 | hdmapval2 41815 | . 2 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), (𝑋 + 𝑌)〉)) |
52 | 3, 6, 44, 25, 29, 30, 32 | lspindpi 21152 | . . . . . 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 41757 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑋}))) |
55 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 29, 25, 54 | hdmapval2 41815 | . . 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 41757 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑌}))) |
58 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 30, 25, 57 | hdmapval2 41815 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉)) |
59 | 55, 58 | oveq12d 7449 | . 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 2785 | 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 2106 ≠ wne 2938 {csn 4631 {cpr 4633 〈cop 4637 〈cotp 4639 I cid 5582 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 -gcsg 18966 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 LTrncltrn 40084 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607 HVMapchvm 41739 HDMap1chdma1 41774 HDMapchdma 41775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608 df-hvmap 41740 df-hdmap1 41776 df-hdmap 41777 |
This theorem is referenced by: hdmap11lem2 41825 |
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