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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 34439 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 2610 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
2 | dihp.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
3 | eqid 2610 | . 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 35416 | . 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 35642 | . 2 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈)) |
11 | eqid 2610 | . . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾) | |
12 | eqid 2610 | . . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
13 | 11, 12, 4, 8 | lhpocnel2 34323 | . . . . . . . 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 2610 | . . . . . . . 8 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) | |
18 | 15, 11, 12, 4, 16, 17 | ltrniotaidvalN 34889 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
19 | 6, 14, 18 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
20 | 19 | fveq2d 6107 | . . . . 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 35079 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
25 | 6, 22, 24 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
26 | 20, 25 | eqtr2d 2645 | . . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) |
27 | fvex 6113 | . . . . . . 7 ⊢ (Base‘𝐾) ∈ V | |
28 | 15, 27 | eqeltri 2684 | . . . . . 6 ⊢ 𝐵 ∈ V |
29 | resiexg 6994 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
30 | 28, 29 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) |
31 | eqeq1 2614 | . . . . . . 7 ⊢ (𝑔 = ( I ↾ 𝐵) → (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))) | |
32 | 31 | anbi1d 737 | . . . . . 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 6102 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) | |
34 | 33 | eqeq2d 2620 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))) |
35 | eleq1 2676 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸)) | |
36 | 34, 35 | anbi12d 743 | . . . . . 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 4918 | . . . . 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 691 | . . . 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 959 | . . 3 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ∈ {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
40 | eqid 2610 | . . . . . 6 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
41 | 11, 12, 4, 40, 9 | dihvalcqat 35546 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
42 | 6, 14, 41 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
43 | 11, 12, 4, 8, 16, 23, 40 | dicval 35483 | . . . . 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 691 | . . . 4 ⊢ (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
45 | 42, 44 | eqtr2d 2645 | . . 3 ⊢ (𝜑 → {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} = (𝐼‘𝑃)) |
46 | 39, 45 | eleqtrd 2690 | . 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 35418 | . . . . . . 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 2620 | . . . . 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 6113 | . . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V | |
54 | 16, 53 | eqeltri 2684 | . . . . . . . . 9 ⊢ 𝑇 ∈ V |
55 | 54 | mptex 6390 | . . . . . . . 8 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V |
56 | 48, 55 | eqeltri 2684 | . . . . . . 7 ⊢ 𝑂 ∈ V |
57 | 52, 56 | opth2 4875 | . . . . . 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 242 | . . . 4 ⊢ (𝜑 → (〈( I ↾ 𝐵), 𝑆〉 = (0g‘𝑈) → 𝑆 = 𝑂)) |
60 | 59 | necon3d 2803 | . . 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 33310 | 1 ⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{〈( I ↾ 𝐵), 𝑆〉})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 {csn 4125 〈cop 4131 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 I cid 4948 ↾ cres 5040 ‘cfv 5804 ℩crio 6510 Basecbs 15695 lecple 15775 occoc 15776 0gc0g 15923 LSpanclspn 18792 LSAtomsclsa 33279 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 TEndoctendo 35058 DVecHcdvh 35385 DIsoCcdic 35479 DIsoHcdih 35535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tendo 35061 df-edring 35063 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 |
This theorem is referenced by: (None) |
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