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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihpN | Structured version Visualization version GIF version |
Description: The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector 〈0, 𝑆〉 (the zero translation ltrnid 37804 and a nonzero member of the endomorphism ring). In particular, 𝑆 can be replaced with the ring unit ( I ↾ 𝑇). (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihp.b | ⊢ 𝐵 = (Base‘𝐾) |
dihp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihp.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
dihp.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dihp.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dihp.o | ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
dihp.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihp.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihp.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dihp.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihp.s | ⊢ (𝜑 → (𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂)) |
Ref | Expression |
---|---|
dihpN | ⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{〈( I ↾ 𝐵), 𝑆〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2739 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
2 | dihp.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
3 | eqid 2739 | . 2 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
4 | dihp.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dihp.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | dihp.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | dvhlvec 38778 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
8 | dihp.p | . . 3 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
9 | dihp.i | . . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
10 | 4, 8, 9, 5, 3, 6 | dihat 39004 | . 2 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈)) |
11 | eqid 2739 | . . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾) | |
12 | eqid 2739 | . . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
13 | 11, 12, 4, 8 | lhpocnel2 37688 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) |
14 | dihp.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾) | |
15 | dihp.t | . . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
16 | eqid 2739 | . . . . . . . 8 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) | |
17 | 14, 11, 12, 4, 15, 16 | ltrniotaidvalN 38252 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
18 | 6, 13, 17 | syl2anc2 588 | . . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
19 | 18 | fveq2d 6690 | . . . . 5 ⊢ (𝜑 → (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘( I ↾ 𝐵))) |
20 | dihp.s | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂)) | |
21 | 20 | simpld 498 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐸) |
22 | dihp.e | . . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
23 | 14, 4, 22 | tendoid 38442 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
24 | 6, 21, 23 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
25 | 19, 24 | eqtr2d 2775 | . . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) |
26 | 14 | fvexi 6700 | . . . . . 6 ⊢ 𝐵 ∈ V |
27 | resiexg 7657 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
28 | 26, 27 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) |
29 | eqeq1 2743 | . . . . . . 7 ⊢ (𝑔 = ( I ↾ 𝐵) → (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))) | |
30 | 29 | anbi1d 633 | . . . . . 6 ⊢ (𝑔 = ( I ↾ 𝐵) → ((𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸) ↔ (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸))) |
31 | fveq1 6685 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) | |
32 | 31 | eqeq2d 2750 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))) |
33 | eleq1 2821 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸)) | |
34 | 32, 33 | anbi12d 634 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ((( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸) ↔ (( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑆 ∈ 𝐸))) |
35 | 30, 34 | opelopabg 5403 | . . . . 5 ⊢ ((( I ↾ 𝐵) ∈ V ∧ 𝑆 ∈ 𝐸) → (〈( I ↾ 𝐵), 𝑆〉 ∈ {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} ↔ (( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑆 ∈ 𝐸))) |
36 | 28, 21, 35 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (〈( I ↾ 𝐵), 𝑆〉 ∈ {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} ↔ (( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑆 ∈ 𝐸))) |
37 | 25, 21, 36 | mpbir2and 713 | . . 3 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ∈ {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
38 | eqid 2739 | . . . . . 6 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
39 | 11, 12, 4, 38, 9 | dihvalcqat 38908 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
40 | 6, 13, 39 | syl2anc2 588 | . . . 4 ⊢ (𝜑 → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
41 | 11, 12, 4, 8, 15, 22, 38 | dicval 38845 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
42 | 6, 13, 41 | syl2anc2 588 | . . . 4 ⊢ (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
43 | 40, 42 | eqtr2d 2775 | . . 3 ⊢ (𝜑 → {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} = (𝐼‘𝑃)) |
44 | 37, 43 | eleqtrd 2836 | . 2 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ∈ (𝐼‘𝑃)) |
45 | 20 | simprd 499 | . . 3 ⊢ (𝜑 → 𝑆 ≠ 𝑂) |
46 | dihp.o | . . . . . . . 8 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
47 | 14, 4, 15, 5, 1, 46 | dvh0g 38780 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0g‘𝑈) = 〈( I ↾ 𝐵), 𝑂〉) |
48 | 6, 47 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑈) = 〈( I ↾ 𝐵), 𝑂〉) |
49 | 48 | eqeq2d 2750 | . . . . 5 ⊢ (𝜑 → (〈( I ↾ 𝐵), 𝑆〉 = (0g‘𝑈) ↔ 〈( I ↾ 𝐵), 𝑆〉 = 〈( I ↾ 𝐵), 𝑂〉)) |
50 | 26, 27 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ 𝐵) ∈ V |
51 | 15 | fvexi 6700 | . . . . . . . . 9 ⊢ 𝑇 ∈ V |
52 | 51 | mptex 7008 | . . . . . . . 8 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V |
53 | 46, 52 | eqeltri 2830 | . . . . . . 7 ⊢ 𝑂 ∈ V |
54 | 50, 53 | opth2 5348 | . . . . . 6 ⊢ (〈( I ↾ 𝐵), 𝑆〉 = 〈( I ↾ 𝐵), 𝑂〉 ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ 𝑆 = 𝑂)) |
55 | 54 | simprbi 500 | . . . . 5 ⊢ (〈( I ↾ 𝐵), 𝑆〉 = 〈( I ↾ 𝐵), 𝑂〉 → 𝑆 = 𝑂) |
56 | 49, 55 | syl6bi 256 | . . . 4 ⊢ (𝜑 → (〈( I ↾ 𝐵), 𝑆〉 = (0g‘𝑈) → 𝑆 = 𝑂)) |
57 | 56 | necon3d 2956 | . . 3 ⊢ (𝜑 → (𝑆 ≠ 𝑂 → 〈( I ↾ 𝐵), 𝑆〉 ≠ (0g‘𝑈))) |
58 | 45, 57 | mpd 15 | . 2 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ≠ (0g‘𝑈)) |
59 | 1, 2, 3, 7, 10, 44, 58 | lsatel 36674 | 1 ⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{〈( I ↾ 𝐵), 𝑆〉})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 Vcvv 3400 {csn 4526 〈cop 4532 class class class wbr 5040 {copab 5102 ↦ cmpt 5120 I cid 5438 ↾ cres 5537 ‘cfv 6349 ℩crio 7138 Basecbs 16598 lecple 16687 occoc 16688 0gc0g 16828 LSpanclspn 19874 LSAtomsclsa 36643 Atomscatm 36932 HLchlt 37019 LHypclh 37653 LTrncltrn 37770 TEndoctendo 38421 DVecHcdvh 38747 DIsoCcdic 38841 DIsoHcdih 38897 |
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 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-riotaBAD 36622 |
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 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-tpos 7933 df-undef 7980 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-n0 11989 df-z 12075 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-sca 16696 df-vsca 16697 df-0g 16830 df-proset 17666 df-poset 17684 df-plt 17696 df-lub 17712 df-glb 17713 df-join 17714 df-meet 17715 df-p0 17777 df-p1 17778 df-lat 17784 df-clat 17846 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-submnd 18085 df-grp 18234 df-minusg 18235 df-sbg 18236 df-subg 18406 df-cntz 18577 df-lsm 18891 df-cmn 19038 df-abl 19039 df-mgp 19371 df-ur 19383 df-ring 19430 df-oppr 19507 df-dvdsr 19525 df-unit 19526 df-invr 19556 df-dvr 19567 df-drng 19635 df-lmod 19767 df-lss 19835 df-lsp 19875 df-lvec 20006 df-lsatoms 36645 df-oposet 36845 df-ol 36847 df-oml 36848 df-covers 36935 df-ats 36936 df-atl 36967 df-cvlat 36991 df-hlat 37020 df-llines 37167 df-lplanes 37168 df-lvols 37169 df-lines 37170 df-psubsp 37172 df-pmap 37173 df-padd 37465 df-lhyp 37657 df-laut 37658 df-ldil 37773 df-ltrn 37774 df-trl 37828 df-tendo 38424 df-edring 38426 df-disoa 38698 df-dvech 38748 df-dib 38808 df-dic 38842 df-dih 38898 |
This theorem is referenced by: (None) |
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