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Mathbox for Norm Megill |
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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 35739 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 2651 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
2 | dihp.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
3 | eqid 2651 | . 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 36715 | . 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 36941 | . 2 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈)) |
11 | eqid 2651 | . . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾) | |
12 | eqid 2651 | . . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
13 | 11, 12, 4, 8 | lhpocnel2 35623 | . . . . . . . 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 2651 | . . . . . . . 8 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) | |
18 | 15, 11, 12, 4, 16, 17 | ltrniotaidvalN 36188 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
19 | 6, 14, 18 | syl2anc 694 | . . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
20 | 19 | fveq2d 6233 | . . . . 5 ⊢ (𝜑 → (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘( I ↾ 𝐵))) |
21 | dihp.s | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂)) | |
22 | 21 | simpld 474 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐸) |
23 | dihp.e | . . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
24 | 15, 4, 23 | tendoid 36378 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
25 | 6, 22, 24 | syl2anc 694 | . . . . 5 ⊢ (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
26 | 20, 25 | eqtr2d 2686 | . . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) |
27 | fvex 6239 | . . . . . . 7 ⊢ (Base‘𝐾) ∈ V | |
28 | 15, 27 | eqeltri 2726 | . . . . . 6 ⊢ 𝐵 ∈ V |
29 | resiexg 7144 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
30 | 28, 29 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) |
31 | eqeq1 2655 | . . . . . . 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 6228 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) | |
34 | 33 | eqeq2d 2661 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))) |
35 | eleq1 2718 | . . . . . . 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 5022 | . . . . 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 694 | . . . 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 977 | . . 3 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ∈ {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
40 | eqid 2651 | . . . . . 6 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
41 | 11, 12, 4, 40, 9 | dihvalcqat 36845 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
42 | 6, 14, 41 | syl2anc 694 | . . . 4 ⊢ (𝜑 → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
43 | 11, 12, 4, 8, 16, 23, 40 | dicval 36782 | . . . . 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 694 | . . . 4 ⊢ (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
45 | 42, 44 | eqtr2d 2686 | . . 3 ⊢ (𝜑 → {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} = (𝐼‘𝑃)) |
46 | 39, 45 | eleqtrd 2732 | . 2 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ∈ (𝐼‘𝑃)) |
47 | 21 | simprd 478 | . . 3 ⊢ (𝜑 → 𝑆 ≠ 𝑂) |
48 | dihp.o | . . . . . . . 8 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
49 | 15, 4, 16, 5, 1, 48 | dvh0g 36717 | . . . . . . 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 2661 | . . . . 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 6239 | . . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V | |
54 | 16, 53 | eqeltri 2726 | . . . . . . . . 9 ⊢ 𝑇 ∈ V |
55 | 54 | mptex 6527 | . . . . . . . 8 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V |
56 | 48, 55 | eqeltri 2726 | . . . . . . 7 ⊢ 𝑂 ∈ V |
57 | 52, 56 | opth2 4978 | . . . . . 6 ⊢ (〈( I ↾ 𝐵), 𝑆〉 = 〈( I ↾ 𝐵), 𝑂〉 ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ 𝑆 = 𝑂)) |
58 | 57 | simprbi 479 | . . . . 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 2844 | . . 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 34610 | 1 ⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{〈( I ↾ 𝐵), 𝑆〉})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 Vcvv 3231 {csn 4210 〈cop 4216 class class class wbr 4685 {copab 4745 ↦ cmpt 4762 I cid 5052 ↾ cres 5145 ‘cfv 5926 ℩crio 6650 Basecbs 15904 lecple 15995 occoc 15996 0gc0g 16147 LSpanclspn 19019 LSAtomsclsa 34579 Atomscatm 34868 HLchlt 34955 LHypclh 35588 LTrncltrn 35705 TEndoctendo 36357 DVecHcdvh 36684 DIsoCcdic 36778 DIsoHcdih 36834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-cntz 17796 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lsatoms 34581 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tendo 36360 df-edring 36362 df-disoa 36635 df-dvech 36685 df-dib 36745 df-dic 36779 df-dih 36835 |
This theorem is referenced by: (None) |
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