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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem8 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40893. Baer p. 45, line 4: "...so that (F(x-y))* <= (Fy)*. This would imply that F(x-y) <= F(y)..." (Contributed by NM, 20-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 |
---|---|
mapdpglem8 | ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem4.jt | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
2 | eqid 2731 | . . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
3 | mapdpglem2.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
4 | mapdpglem.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | mapdpglem.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | mapdpglem.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | lcdlmod 40779 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
8 | mapdpglem.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
9 | mapdpglem.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2731 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
11 | 4, 9, 6 | dvhlmod 40297 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | mapdpglem.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
13 | mapdpglem.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
14 | mapdpglem.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈) | |
15 | 13, 10, 14 | lspsncl 20736 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
16 | 11, 12, 15 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
17 | 4, 8, 9, 10, 5, 2, 6, 16 | mapdcl2 40843 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
18 | mapdpglem.s | . . . . 5 ⊢ − = (-g‘𝑈) | |
19 | mapdpglem.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
20 | mapdpglem1.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐶) | |
21 | mapdpglem3.f | . . . . 5 ⊢ 𝐹 = (Base‘𝐶) | |
22 | mapdpglem3.te | . . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
23 | mapdpglem3.a | . . . . 5 ⊢ 𝐴 = (Scalar‘𝑈) | |
24 | mapdpglem3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
25 | mapdpglem3.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐶) | |
26 | mapdpglem3.r | . . . . 5 ⊢ 𝑅 = (-g‘𝐶) | |
27 | mapdpglem3.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
28 | mapdpglem3.e | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
29 | mapdpglem4.q | . . . . 5 ⊢ 𝑄 = (0g‘𝑈) | |
30 | mapdpglem.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
31 | mapdpglem4.z | . . . . 5 ⊢ 0 = (0g‘𝐴) | |
32 | mapdpglem4.g4 | . . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
33 | mapdpglem4.z4 | . . . . 5 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
34 | mapdpglem4.t4 | . . . . 5 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
35 | mapdpglem4.xn | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
36 | mapdpglem4.g0 | . . . . 5 ⊢ (𝜑 → 𝑔 = 0 ) | |
37 | 4, 8, 9, 13, 18, 14, 5, 6, 19, 12, 20, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 1, 31, 32, 33, 34, 35, 36 | mapdpglem6 40865 | . . . 4 ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌}))) |
38 | 2, 3, 7, 17, 37 | lspsnel5a 20755 | . . 3 ⊢ (𝜑 → (𝐽‘{𝑡}) ⊆ (𝑀‘(𝑁‘{𝑌}))) |
39 | 1, 38 | eqsstrd 4020 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑌}))) |
40 | 13, 18 | lmodvsubcl 20665 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
41 | 11, 19, 12, 40 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉) |
42 | 13, 10, 14 | lspsncl 20736 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈)) |
43 | 11, 41, 42 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈)) |
44 | 4, 9, 10, 8, 6, 43, 16 | mapdord 40825 | . 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌}))) |
45 | 39, 44 | mpbid 231 | 1 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ⊆ wss 3948 {csn 4628 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 Scalarcsca 17207 ·𝑠 cvsca 17208 0gc0g 17392 -gcsg 18860 LSSumclsm 19547 LModclmod 20618 LSubSpclss 20690 LSpanclspn 20730 HLchlt 38536 LHypclh 39171 DVecHcdvh 40265 LCDualclcd 40773 mapdcmpd 40811 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38139 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-proset 18255 df-poset 18273 df-plt 18290 df-lub 18306 df-glb 18307 df-join 18308 df-meet 18309 df-p0 18385 df-p1 18386 df-lat 18392 df-clat 18459 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-subg 19043 df-cntz 19226 df-oppg 19255 df-lsm 19549 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-oppr 20229 df-dvdsr 20252 df-unit 20253 df-invr 20283 df-dvr 20296 df-drng 20506 df-lmod 20620 df-lss 20691 df-lsp 20731 df-lvec 20862 df-lsatoms 38162 df-lshyp 38163 df-lcv 38205 df-lfl 38244 df-lkr 38272 df-ldual 38310 df-oposet 38362 df-ol 38364 df-oml 38365 df-covers 38452 df-ats 38453 df-atl 38484 df-cvlat 38508 df-hlat 38537 df-llines 38685 df-lplanes 38686 df-lvols 38687 df-lines 38688 df-psubsp 38690 df-pmap 38691 df-padd 38983 df-lhyp 39175 df-laut 39176 df-ldil 39291 df-ltrn 39292 df-trl 39346 df-tgrp 39930 df-tendo 39942 df-edring 39944 df-dveca 40190 df-disoa 40216 df-dvech 40266 df-dib 40326 df-dic 40360 df-dih 40416 df-doch 40535 df-djh 40582 df-lcdual 40774 df-mapd 40812 |
This theorem is referenced by: mapdpglem9 40867 |
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