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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem6 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 41612. Baer p. 45, line 4: "If g were 0, then t would be in (Fy)*..." (Contributed by NM, 18-Mar-2015.) |
Ref | Expression |
---|---|
mapdpglem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdpglem.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdpglem.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdpglem.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdpglem.s | ⊢ − = (-g‘𝑈) |
mapdpglem.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdpglem.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdpglem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdpglem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
mapdpglem.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
mapdpglem1.p | ⊢ ⊕ = (LSSum‘𝐶) |
mapdpglem2.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdpglem3.f | ⊢ 𝐹 = (Base‘𝐶) |
mapdpglem3.te | ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
mapdpglem3.a | ⊢ 𝐴 = (Scalar‘𝑈) |
mapdpglem3.b | ⊢ 𝐵 = (Base‘𝐴) |
mapdpglem3.t | ⊢ · = ( ·𝑠 ‘𝐶) |
mapdpglem3.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdpglem3.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
mapdpglem3.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) |
mapdpglem4.q | ⊢ 𝑄 = (0g‘𝑈) |
mapdpglem.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdpglem4.jt | ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) |
mapdpglem4.z | ⊢ 0 = (0g‘𝐴) |
mapdpglem4.g4 | ⊢ (𝜑 → 𝑔 ∈ 𝐵) |
mapdpglem4.z4 | ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) |
mapdpglem4.t4 | ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
mapdpglem4.xn | ⊢ (𝜑 → 𝑋 ≠ 𝑄) |
mapdpglem4.g0 | ⊢ (𝜑 → 𝑔 = 0 ) |
Ref | Expression |
---|---|
mapdpglem6 | ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem4.t4 | . 2 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
2 | mapdpglem.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | mapdpglem.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
4 | mapdpglem.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
5 | 2, 3, 4 | lcdlmod 41498 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LMod) |
6 | mapdpglem.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
7 | mapdpglem.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | eqid 2734 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
9 | eqid 2734 | . . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
10 | 2, 7, 4 | dvhlmod 41016 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
11 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
12 | mapdpglem.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
13 | mapdpglem.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
14 | 12, 8, 13 | lspsncl 20993 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
15 | 10, 11, 14 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
16 | 2, 6, 7, 8, 3, 9, 4, 15 | mapdcl2 41562 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
17 | mapdpglem4.g0 | . . . . . 6 ⊢ (𝜑 → 𝑔 = 0 ) | |
18 | 17 | oveq1d 7460 | . . . . 5 ⊢ (𝜑 → (𝑔 · 𝐺) = ( 0 · 𝐺)) |
19 | mapdpglem3.a | . . . . . 6 ⊢ 𝐴 = (Scalar‘𝑈) | |
20 | mapdpglem4.z | . . . . . 6 ⊢ 0 = (0g‘𝐴) | |
21 | mapdpglem3.f | . . . . . 6 ⊢ 𝐹 = (Base‘𝐶) | |
22 | mapdpglem3.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝐶) | |
23 | eqid 2734 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
24 | mapdpglem3.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
25 | 2, 7, 19, 20, 3, 21, 22, 23, 4, 24 | lcd0vs 41521 | . . . . 5 ⊢ (𝜑 → ( 0 · 𝐺) = (0g‘𝐶)) |
26 | 18, 25 | eqtrd 2774 | . . . 4 ⊢ (𝜑 → (𝑔 · 𝐺) = (0g‘𝐶)) |
27 | 23, 9 | lss0cl 20963 | . . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) → (0g‘𝐶) ∈ (𝑀‘(𝑁‘{𝑌}))) |
28 | 5, 16, 27 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (0g‘𝐶) ∈ (𝑀‘(𝑁‘{𝑌}))) |
29 | 26, 28 | eqeltrd 2838 | . . 3 ⊢ (𝜑 → (𝑔 · 𝐺) ∈ (𝑀‘(𝑁‘{𝑌}))) |
30 | mapdpglem4.z4 | . . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
31 | mapdpglem3.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
32 | 31, 9 | lssvsubcl 20960 | . . 3 ⊢ (((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) ∧ ((𝑔 · 𝐺) ∈ (𝑀‘(𝑁‘{𝑌})) ∧ 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))) → ((𝑔 · 𝐺)𝑅𝑧) ∈ (𝑀‘(𝑁‘{𝑌}))) |
33 | 5, 16, 29, 30, 32 | syl22anc 838 | . 2 ⊢ (𝜑 → ((𝑔 · 𝐺)𝑅𝑧) ∈ (𝑀‘(𝑁‘{𝑌}))) |
34 | 1, 33 | eqeltrd 2838 | 1 ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 {csn 4648 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 Scalarcsca 17309 ·𝑠 cvsca 17310 0gc0g 17494 -gcsg 18970 LSSumclsm 19671 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 HLchlt 39255 LHypclh 39890 DVecHcdvh 40984 LCDualclcd 41492 mapdcmpd 41530 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-riotaBAD 38858 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-tpos 8263 df-undef 8310 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-0g 17496 df-mre 17639 df-mrc 17640 df-acs 17642 df-proset 18360 df-poset 18378 df-plt 18395 df-lub 18411 df-glb 18412 df-join 18413 df-meet 18414 df-p0 18490 df-p1 18491 df-lat 18497 df-clat 18564 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-grp 18971 df-minusg 18972 df-sbg 18973 df-subg 19158 df-cntz 19352 df-oppg 19381 df-lsm 19673 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-nzr 20534 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38881 df-lshyp 38882 df-lcv 38924 df-lfl 38963 df-lkr 38991 df-ldual 39029 df-oposet 39081 df-ol 39083 df-oml 39084 df-covers 39171 df-ats 39172 df-atl 39203 df-cvlat 39227 df-hlat 39256 df-llines 39404 df-lplanes 39405 df-lvols 39406 df-lines 39407 df-psubsp 39409 df-pmap 39410 df-padd 39702 df-lhyp 39894 df-laut 39895 df-ldil 40010 df-ltrn 40011 df-trl 40065 df-tgrp 40649 df-tendo 40661 df-edring 40663 df-dveca 40909 df-disoa 40935 df-dvech 40985 df-dib 41045 df-dic 41079 df-dih 41135 df-doch 41254 df-djh 41301 df-lcdual 41493 df-mapd 41531 |
This theorem is referenced by: mapdpglem8 41585 |
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