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Mirrors > Home > MPE Home > Th. List > Mathboxes > hgmaprnlem2N | Structured version Visualization version GIF version |
Description: Lemma for hgmaprnN 39560. Part 15 of [Baer] p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero 𝑧 is taken care of automatically. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hgmaprnlem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hgmaprnlem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hgmaprnlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
hgmaprnlem1.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hgmaprnlem1.b | ⊢ 𝐵 = (Base‘𝑅) |
hgmaprnlem1.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hgmaprnlem1.o | ⊢ 0 = (0g‘𝑈) |
hgmaprnlem1.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hgmaprnlem1.d | ⊢ 𝐷 = (Base‘𝐶) |
hgmaprnlem1.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hgmaprnlem1.a | ⊢ 𝐴 = (Base‘𝑃) |
hgmaprnlem1.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hgmaprnlem1.q | ⊢ 𝑄 = (0g‘𝐶) |
hgmaprnlem1.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hgmaprnlem1.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hgmaprnlem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hgmaprnlem1.z | ⊢ (𝜑 → 𝑧 ∈ 𝐴) |
hgmaprnlem1.t2 | ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) |
hgmaprnlem1.s2 | ⊢ (𝜑 → 𝑠 ∈ 𝑉) |
hgmaprnlem1.sz | ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) |
hgmaprnlem1.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hgmaprnlem1.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hgmaprnlem1.l | ⊢ 𝐿 = (LSpan‘𝐶) |
Ref | Expression |
---|---|
hgmaprnlem2N | ⊢ (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hgmaprnlem1.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hgmaprnlem1.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
3 | hgmaprnlem1.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 39251 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | hgmaprnlem1.z | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴) | |
6 | hgmaprnlem1.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | hgmaprnlem1.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
8 | hgmaprnlem1.d | . . . . 5 ⊢ 𝐷 = (Base‘𝐶) | |
9 | hgmaprnlem1.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
10 | hgmaprnlem1.t2 | . . . . . 6 ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) | |
11 | 10 | eldifad 3855 | . . . . 5 ⊢ (𝜑 → 𝑡 ∈ 𝑉) |
12 | 1, 6, 7, 2, 8, 9, 3, 11 | hdmapcl 39489 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷) |
13 | hgmaprnlem1.p | . . . . 5 ⊢ 𝑃 = (Scalar‘𝐶) | |
14 | hgmaprnlem1.a | . . . . 5 ⊢ 𝐴 = (Base‘𝑃) | |
15 | hgmaprnlem1.e | . . . . 5 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
16 | hgmaprnlem1.l | . . . . 5 ⊢ 𝐿 = (LSpan‘𝐶) | |
17 | 13, 14, 8, 15, 16 | lspsnvsi 19897 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))}) ⊆ (𝐿‘{(𝑆‘𝑡)})) |
18 | 4, 5, 12, 17 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))}) ⊆ (𝐿‘{(𝑆‘𝑡)})) |
19 | hgmaprnlem1.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
20 | hgmaprnlem1.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
21 | hgmaprnlem1.s2 | . . . . 5 ⊢ (𝜑 → 𝑠 ∈ 𝑉) | |
22 | 1, 6, 7, 19, 2, 16, 20, 9, 3, 21 | hdmap10 39499 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑠})) = (𝐿‘{(𝑆‘𝑠)})) |
23 | hgmaprnlem1.sz | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) | |
24 | 23 | sneqd 4528 | . . . . 5 ⊢ (𝜑 → {(𝑆‘𝑠)} = {(𝑧 ∙ (𝑆‘𝑡))}) |
25 | 24 | fveq2d 6680 | . . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑠)}) = (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))})) |
26 | 22, 25 | eqtrd 2773 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑠})) = (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))})) |
27 | 1, 6, 7, 19, 2, 16, 20, 9, 3, 11 | hdmap10 39499 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{(𝑆‘𝑡)})) |
28 | 18, 26, 27 | 3sstr4d 3924 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑠})) ⊆ (𝑀‘(𝑁‘{𝑡}))) |
29 | eqid 2738 | . . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
30 | 1, 6, 3 | dvhlmod 38769 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
31 | 7, 29, 19 | lspsncl 19870 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑠 ∈ 𝑉) → (𝑁‘{𝑠}) ∈ (LSubSp‘𝑈)) |
32 | 30, 21, 31 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑠}) ∈ (LSubSp‘𝑈)) |
33 | 7, 29, 19 | lspsncl 19870 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉) → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈)) |
34 | 30, 11, 33 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈)) |
35 | 1, 6, 29, 20, 3, 32, 34 | mapdord 39297 | . 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑠})) ⊆ (𝑀‘(𝑁‘{𝑡})) ↔ (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡}))) |
36 | 28, 35 | mpbid 235 | 1 ⊢ (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∖ cdif 3840 ⊆ wss 3843 {csn 4516 ‘cfv 6339 (class class class)co 7172 Basecbs 16588 Scalarcsca 16673 ·𝑠 cvsca 16674 0gc0g 16818 LModclmod 19755 LSubSpclss 19824 LSpanclspn 19864 HLchlt 37009 LHypclh 37643 DVecHcdvh 38737 LCDualclcd 39245 mapdcmpd 39283 HDMapchdma 39451 HGMapchg 39542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-riotaBAD 36612 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-ot 4525 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-om 7602 df-1st 7716 df-2nd 7717 df-tpos 7923 df-undef 7970 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-map 8441 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-n0 11979 df-z 12065 df-uz 12327 df-fz 12984 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-mulr 16684 df-sca 16686 df-vsca 16687 df-0g 16820 df-mre 16962 df-mrc 16963 df-acs 16965 df-proset 17656 df-poset 17674 df-plt 17686 df-lub 17702 df-glb 17703 df-join 17704 df-meet 17705 df-p0 17767 df-p1 17768 df-lat 17774 df-clat 17836 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-submnd 18075 df-grp 18224 df-minusg 18225 df-sbg 18226 df-subg 18396 df-cntz 18567 df-oppg 18594 df-lsm 18881 df-cmn 19028 df-abl 19029 df-mgp 19361 df-ur 19373 df-ring 19420 df-oppr 19497 df-dvdsr 19515 df-unit 19516 df-invr 19546 df-dvr 19557 df-drng 19625 df-lmod 19757 df-lss 19825 df-lsp 19865 df-lvec 19996 df-lsatoms 36635 df-lshyp 36636 df-lcv 36678 df-lfl 36717 df-lkr 36745 df-ldual 36783 df-oposet 36835 df-ol 36837 df-oml 36838 df-covers 36925 df-ats 36926 df-atl 36957 df-cvlat 36981 df-hlat 37010 df-llines 37157 df-lplanes 37158 df-lvols 37159 df-lines 37160 df-psubsp 37162 df-pmap 37163 df-padd 37455 df-lhyp 37647 df-laut 37648 df-ldil 37763 df-ltrn 37764 df-trl 37818 df-tgrp 38402 df-tendo 38414 df-edring 38416 df-dveca 38662 df-disoa 38688 df-dvech 38738 df-dib 38798 df-dic 38832 df-dih 38888 df-doch 39007 df-djh 39054 df-lcdual 39246 df-mapd 39284 df-hvmap 39416 df-hdmap1 39452 df-hdmap 39453 |
This theorem is referenced by: hgmaprnlem3N 39557 |
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