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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 35421 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 2622 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
2 | dihp.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
3 | eqid 2622 | . 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 36398 | . 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 36624 | . 2 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈)) |
11 | eqid 2622 | . . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾) | |
12 | eqid 2622 | . . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
13 | 11, 12, 4, 8 | lhpocnel2 35305 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) |
14 | 6, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) |
15 | dihp.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾) | |
16 | dihp.t | . . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
17 | eqid 2622 | . . . . . . . 8 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) | |
18 | 15, 11, 12, 4, 16, 17 | ltrniotaidvalN 35871 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
19 | 6, 14, 18 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
20 | 19 | fveq2d 6195 | . . . . 5 ⊢ (𝜑 → (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘( I ↾ 𝐵))) |
21 | dihp.s | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂)) | |
22 | 21 | simpld 475 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐸) |
23 | dihp.e | . . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
24 | 15, 4, 23 | tendoid 36061 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
25 | 6, 22, 24 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
26 | 20, 25 | eqtr2d 2657 | . . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) |
27 | fvex 6201 | . . . . . . 7 ⊢ (Base‘𝐾) ∈ V | |
28 | 15, 27 | eqeltri 2697 | . . . . . 6 ⊢ 𝐵 ∈ V |
29 | resiexg 7102 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
30 | 28, 29 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) |
31 | eqeq1 2626 | . . . . . . 7 ⊢ (𝑔 = ( I ↾ 𝐵) → (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))) | |
32 | 31 | anbi1d 741 | . . . . . 6 ⊢ (𝑔 = ( I ↾ 𝐵) → ((𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸) ↔ (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸))) |
33 | fveq1 6190 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) | |
34 | 33 | eqeq2d 2632 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))) |
35 | eleq1 2689 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸)) | |
36 | 34, 35 | anbi12d 747 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ((( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸) ↔ (( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑆 ∈ 𝐸))) |
37 | 32, 36 | opelopabg 4993 | . . . . 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))) |
38 | 30, 22, 37 | syl2anc 693 | . . . 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))) |
39 | 26, 22, 38 | mpbir2and 957 | . . 3 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ∈ {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
40 | eqid 2622 | . . . . . 6 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
41 | 11, 12, 4, 40, 9 | dihvalcqat 36528 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
42 | 6, 14, 41 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
43 | 11, 12, 4, 8, 16, 23, 40 | dicval 36465 | . . . . 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)}) |
44 | 6, 14, 43 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
45 | 42, 44 | eqtr2d 2657 | . . 3 ⊢ (𝜑 → {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} = (𝐼‘𝑃)) |
46 | 39, 45 | eleqtrd 2703 | . 2 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ∈ (𝐼‘𝑃)) |
47 | 21 | simprd 479 | . . 3 ⊢ (𝜑 → 𝑆 ≠ 𝑂) |
48 | dihp.o | . . . . . . . 8 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
49 | 15, 4, 16, 5, 1, 48 | dvh0g 36400 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0g‘𝑈) = 〈( I ↾ 𝐵), 𝑂〉) |
50 | 6, 49 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑈) = 〈( I ↾ 𝐵), 𝑂〉) |
51 | 50 | eqeq2d 2632 | . . . . 5 ⊢ (𝜑 → (〈( I ↾ 𝐵), 𝑆〉 = (0g‘𝑈) ↔ 〈( I ↾ 𝐵), 𝑆〉 = 〈( I ↾ 𝐵), 𝑂〉)) |
52 | 28, 29 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ 𝐵) ∈ V |
53 | fvex 6201 | . . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V | |
54 | 16, 53 | eqeltri 2697 | . . . . . . . . 9 ⊢ 𝑇 ∈ V |
55 | 54 | mptex 6486 | . . . . . . . 8 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V |
56 | 48, 55 | eqeltri 2697 | . . . . . . 7 ⊢ 𝑂 ∈ V |
57 | 52, 56 | opth2 4949 | . . . . . 6 ⊢ (〈( I ↾ 𝐵), 𝑆〉 = 〈( I ↾ 𝐵), 𝑂〉 ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ 𝑆 = 𝑂)) |
58 | 57 | simprbi 480 | . . . . 5 ⊢ (〈( I ↾ 𝐵), 𝑆〉 = 〈( I ↾ 𝐵), 𝑂〉 → 𝑆 = 𝑂) |
59 | 51, 58 | syl6bi 243 | . . . 4 ⊢ (𝜑 → (〈( I ↾ 𝐵), 𝑆〉 = (0g‘𝑈) → 𝑆 = 𝑂)) |
60 | 59 | necon3d 2815 | . . 3 ⊢ (𝜑 → (𝑆 ≠ 𝑂 → 〈( I ↾ 𝐵), 𝑆〉 ≠ (0g‘𝑈))) |
61 | 47, 60 | mpd 15 | . 2 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ≠ (0g‘𝑈)) |
62 | 1, 2, 3, 7, 10, 46, 61 | lsatel 34292 | 1 ⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{〈( I ↾ 𝐵), 𝑆〉})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 {csn 4177 〈cop 4183 class class class wbr 4653 {copab 4712 ↦ cmpt 4729 I cid 5023 ↾ cres 5116 ‘cfv 5888 ℩crio 6610 Basecbs 15857 lecple 15948 occoc 15949 0gc0g 16100 LSpanclspn 18971 LSAtomsclsa 34261 Atomscatm 34550 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 TEndoctendo 36040 DVecHcdvh 36367 DIsoCcdic 36461 DIsoHcdih 36517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tendo 36043 df-edring 36045 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 |
This theorem is referenced by: (None) |
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