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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapglem7a | Structured version Visualization version GIF version | ||
| Description: Lemma for hdmapg 41932. (Contributed by NM, 14-Jun-2015.) | 
| Ref | Expression | 
|---|---|
| hdmapglem7.h | ⊢ 𝐻 = (LHyp‘𝐾) | 
| hdmapglem7.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | 
| hdmapglem7.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | 
| hdmapglem7.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | 
| hdmapglem7.v | ⊢ 𝑉 = (Base‘𝑈) | 
| hdmapglem7.p | ⊢ + = (+g‘𝑈) | 
| hdmapglem7.q | ⊢ · = ( ·𝑠 ‘𝑈) | 
| hdmapglem7.r | ⊢ 𝑅 = (Scalar‘𝑈) | 
| hdmapglem7.b | ⊢ 𝐵 = (Base‘𝑅) | 
| hdmapglem7.a | ⊢ ⊕ = (LSSum‘𝑈) | 
| hdmapglem7.n | ⊢ 𝑁 = (LSpan‘𝑈) | 
| hdmapglem7.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| hdmapglem7.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) | 
| Ref | Expression | 
|---|---|
| hdmapglem7a | ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hdmapglem7.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 2 | hdmapglem7.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 3 | hdmapglem7.o | . . . . . 6 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
| 4 | hdmapglem7.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 5 | hdmapglem7.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
| 6 | eqid 2737 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
| 7 | hdmapglem7.a | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑈) | |
| 8 | hdmapglem7.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 9 | 2, 4, 8 | dvhlmod 41112 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) | 
| 10 | eqid 2737 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 11 | eqid 2737 | . . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
| 12 | eqid 2737 | . . . . . . . . 9 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
| 13 | hdmapglem7.e | . . . . . . . . 9 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
| 14 | 2, 10, 11, 4, 5, 12, 13, 8 | dvheveccl 41114 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) | 
| 15 | 14 | eldifad 3963 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉) | 
| 16 | hdmapglem7.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 17 | 5, 6, 16 | lspsncl 20975 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈)) | 
| 18 | 9, 15, 17 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈)) | 
| 19 | 15 | snssd 4809 | . . . . . . . . 9 ⊢ (𝜑 → {𝐸} ⊆ 𝑉) | 
| 20 | 2, 4, 3, 5, 16, 8, 19 | dochocsp 41381 | . . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑁‘{𝐸})) = (𝑂‘{𝐸})) | 
| 21 | 20 | fveq2d 6910 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑂‘(𝑂‘{𝐸}))) | 
| 22 | 2, 4, 3, 5, 16, 8, 15 | dochocsn 41383 | . . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘{𝐸})) = (𝑁‘{𝐸})) | 
| 23 | 21, 22 | eqtrd 2777 | . . . . . 6 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑁‘{𝐸})) | 
| 24 | 2, 3, 4, 5, 6, 7, 8, 18, 23 | dochexmid 41470 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = 𝑉) | 
| 25 | 20 | oveq2d 7447 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) | 
| 26 | 24, 25 | eqtr3d 2779 | . . . 4 ⊢ (𝜑 → 𝑉 = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) | 
| 27 | 1, 26 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸}))) | 
| 28 | 6 | lsssssubg 20956 | . . . . . 6 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) | 
| 29 | 9, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈)) | 
| 30 | 29, 18 | sseldd 3984 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (SubGrp‘𝑈)) | 
| 31 | 2, 4, 5, 6, 3 | dochlss 41356 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈)) | 
| 32 | 8, 19, 31 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈)) | 
| 33 | 29, 32 | sseldd 3984 | . . . 4 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈)) | 
| 34 | hdmapglem7.p | . . . . 5 ⊢ + = (+g‘𝑈) | |
| 35 | 34, 7 | lsmelval 19667 | . . . 4 ⊢ (((𝑁‘{𝐸}) ∈ (SubGrp‘𝑈) ∧ (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈)) → (𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})) ↔ ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢))) | 
| 36 | 30, 33, 35 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})) ↔ ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢))) | 
| 37 | 27, 36 | mpbid 232 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢)) | 
| 38 | rexcom 3290 | . . 3 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢)) | |
| 39 | df-rex 3071 | . . . . 5 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢))) | |
| 40 | hdmapglem7.r | . . . . . . . . . . 11 ⊢ 𝑅 = (Scalar‘𝑈) | |
| 41 | hdmapglem7.b | . . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅) | |
| 42 | hdmapglem7.q | . . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝑈) | |
| 43 | 40, 41, 5, 42, 16 | ellspsn 21001 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑎 ∈ (𝑁‘{𝐸}) ↔ ∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸))) | 
| 44 | 9, 15, 43 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝑎 ∈ (𝑁‘{𝐸}) ↔ ∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸))) | 
| 45 | 44 | anbi1d 631 | . . . . . . . 8 ⊢ (𝜑 → ((𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ (∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))) | 
| 46 | r19.41v 3189 | . . . . . . . 8 ⊢ (∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ (∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))) | |
| 47 | 45, 46 | bitr4di 289 | . . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))) | 
| 48 | 47 | exbidv 1921 | . . . . . 6 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))) | 
| 49 | rexcom4 3288 | . . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))) | |
| 50 | ovex 7464 | . . . . . . . . 9 ⊢ (𝑘 · 𝐸) ∈ V | |
| 51 | oveq1 7438 | . . . . . . . . . 10 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑎 + 𝑢) = ((𝑘 · 𝐸) + 𝑢)) | |
| 52 | 51 | eqeq2d 2748 | . . . . . . . . 9 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑋 = (𝑎 + 𝑢) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢))) | 
| 53 | 50, 52 | ceqsexv 3532 | . . . . . . . 8 ⊢ (∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢)) | 
| 54 | 53 | rexbii 3094 | . . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) | 
| 55 | 49, 54 | bitr3i 277 | . . . . . 6 ⊢ (∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) | 
| 56 | 48, 55 | bitrdi 287 | . . . . 5 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) | 
| 57 | 39, 56 | bitrid 283 | . . . 4 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) | 
| 58 | 57 | rexbidv 3179 | . . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) | 
| 59 | 38, 58 | bitrid 283 | . 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))) | 
| 60 | 37, 59 | mpbid 232 | 1 ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 {csn 4626 〈cop 4632 I cid 5577 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 SubGrpcsubg 19138 LSSumclsm 19652 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 DVecHcdvh 41080 ocHcoch 41349 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lcv 39020 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tgrp 40745 df-tendo 40757 df-edring 40759 df-dveca 41005 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 df-doch 41350 df-djh 41397 | 
| This theorem is referenced by: hdmapglem7 41931 | 
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