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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for mapdpg 37860. Baer p. 44, last line: "(F(x-y))* <= (Fx)*+(Fy)*." (Contributed by NM, 15-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) |
| Ref | Expression |
|---|---|
| mapdpglem1 | ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapdpglem.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 2 | mapdpglem.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 3 | mapdpglem.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 4 | 1, 2, 3 | dvhlmod 37264 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 5 | mapdpglem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 6 | mapdpglem.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
| 7 | mapdpglem.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
| 8 | mapdpglem.s | . . . . 5 ⊢ − = (-g‘𝑈) | |
| 9 | eqid 2778 | . . . . 5 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
| 10 | mapdpglem.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 11 | 7, 8, 9, 10 | lspsntrim 19493 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌}))) |
| 12 | 4, 5, 6, 11 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌}))) |
| 13 | eqid 2778 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
| 14 | mapdpglem.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
| 15 | 7, 8 | lmodvsubcl 19300 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
| 16 | 4, 5, 6, 15 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉) |
| 17 | 7, 13, 10 | lspsncl 19372 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈)) |
| 18 | 4, 16, 17 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈)) |
| 19 | 7, 13, 10 | lspsncl 19372 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
| 20 | 4, 5, 19 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
| 21 | 7, 13, 10 | lspsncl 19372 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
| 22 | 4, 6, 21 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
| 23 | 13, 9 | lsmcl 19478 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑈)) |
| 24 | 4, 20, 22, 23 | syl3anc 1439 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑈)) |
| 25 | 1, 2, 13, 14, 3, 18, 24 | mapdord 37792 | . . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌}))) ↔ (𝑁‘{(𝑋 − 𝑌)}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))) |
| 26 | 12, 25 | mpbird 249 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))) |
| 27 | mapdpglem.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
| 28 | mapdpglem1.p | . . 3 ⊢ ⊕ = (LSSum‘𝐶) | |
| 29 | 1, 14, 2, 13, 9, 27, 28, 3, 20, 22 | mapdlsm 37818 | . 2 ⊢ (𝜑 → (𝑀‘((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌}))) = ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
| 30 | 26, 29 | sseqtrd 3860 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 {csn 4398 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 -gcsg 17811 LSSumclsm 18433 LModclmod 19255 LSubSpclss 19324 LSpanclspn 19366 HLchlt 35504 LHypclh 36138 DVecHcdvh 37232 LCDualclcd 37740 mapdcmpd 37778 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35107 |
| This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-mre 16632 df-mrc 16633 df-acs 16635 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-oppg 18159 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35130 df-lshyp 35131 df-lcv 35173 df-lfl 35212 df-lkr 35240 df-ldual 35278 df-oposet 35330 df-ol 35332 df-oml 35333 df-covers 35420 df-ats 35421 df-atl 35452 df-cvlat 35476 df-hlat 35505 df-llines 35652 df-lplanes 35653 df-lvols 35654 df-lines 35655 df-psubsp 35657 df-pmap 35658 df-padd 35950 df-lhyp 36142 df-laut 36143 df-ldil 36258 df-ltrn 36259 df-trl 36313 df-tgrp 36897 df-tendo 36909 df-edring 36911 df-dveca 37157 df-disoa 37183 df-dvech 37233 df-dib 37293 df-dic 37327 df-dih 37383 df-doch 37502 df-djh 37549 df-lcdual 37741 df-mapd 37779 |
| This theorem is referenced by: mapdpglem2 37827 |
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