| 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 40137 and a nonzero member of the endomorphism ring). In particular, 𝑆 can be replaced with the ring unity ( 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 2737 | . 2 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
| 2 | dihp.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 3 | eqid 2737 | . 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 41111 | . 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 41337 | . 2 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈)) |
| 11 | eqid 2737 | . . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 12 | eqid 2737 | . . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
| 13 | 11, 12, 4, 8 | lhpocnel2 40021 | . . . . . . 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 2737 | . . . . . . . 8 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) | |
| 17 | 14, 11, 12, 4, 15, 16 | ltrniotaidvalN 40585 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
| 18 | 6, 13, 17 | syl2anc2 585 | . . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵)) |
| 19 | 18 | fveq2d 6910 | . . . . 5 ⊢ (𝜑 → (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘( I ↾ 𝐵))) |
| 20 | dihp.s | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂)) | |
| 21 | 20 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐸) |
| 22 | dihp.e | . . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
| 23 | 14, 4, 22 | tendoid 40775 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
| 24 | 6, 21, 23 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
| 25 | 19, 24 | eqtr2d 2778 | . . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) |
| 26 | 14 | fvexi 6920 | . . . . . 6 ⊢ 𝐵 ∈ V |
| 27 | resiexg 7934 | . . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V) | |
| 28 | 26, 27 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) |
| 29 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑔 = ( I ↾ 𝐵) → (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))) | |
| 30 | 29 | anbi1d 631 | . . . . . 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 6905 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))) | |
| 32 | 31 | eqeq2d 2748 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))) |
| 33 | eleq1 2829 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸)) | |
| 34 | 32, 33 | anbi12d 632 | . . . . . 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 5543 | . . . . 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 584 | . . . 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 2737 | . . . . . 6 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
| 39 | 11, 12, 4, 38, 9 | dihvalcqat 41241 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
| 40 | 6, 13, 39 | syl2anc2 585 | . . . 4 ⊢ (𝜑 → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃)) |
| 41 | 11, 12, 4, 8, 15, 22, 38 | dicval 41178 | . . . . 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 585 | . . . 4 ⊢ (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)}) |
| 43 | 40, 42 | eqtr2d 2778 | . . 3 ⊢ (𝜑 → {〈𝑔, 𝑠〉 ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} = (𝐼‘𝑃)) |
| 44 | 37, 43 | eleqtrd 2843 | . 2 ⊢ (𝜑 → 〈( I ↾ 𝐵), 𝑆〉 ∈ (𝐼‘𝑃)) |
| 45 | 20 | simprd 495 | . . 3 ⊢ (𝜑 → 𝑆 ≠ 𝑂) |
| 46 | dihp.o | . . . . . . . 8 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
| 47 | 14, 4, 15, 5, 1, 46 | dvh0g 41113 | . . . . . . 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 2748 | . . . . 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 6920 | . . . . . . . . 9 ⊢ 𝑇 ∈ V |
| 52 | 51 | mptex 7243 | . . . . . . . 8 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V |
| 53 | 46, 52 | eqeltri 2837 | . . . . . . 7 ⊢ 𝑂 ∈ V |
| 54 | 50, 53 | opth2 5485 | . . . . . 6 ⊢ (〈( I ↾ 𝐵), 𝑆〉 = 〈( I ↾ 𝐵), 𝑂〉 ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ 𝑆 = 𝑂)) |
| 55 | 54 | simprbi 496 | . . . . 5 ⊢ (〈( I ↾ 𝐵), 𝑆〉 = 〈( I ↾ 𝐵), 𝑂〉 → 𝑆 = 𝑂) |
| 56 | 49, 55 | biimtrdi 253 | . . . 4 ⊢ (𝜑 → (〈( I ↾ 𝐵), 𝑆〉 = (0g‘𝑈) → 𝑆 = 𝑂)) |
| 57 | 56 | necon3d 2961 | . . 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 39006 | 1 ⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{〈( I ↾ 𝐵), 𝑆〉})) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 {csn 4626 〈cop 4632 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 I cid 5577 ↾ cres 5687 ‘cfv 6561 ℩crio 7387 Basecbs 17247 lecple 17304 occoc 17305 0gc0g 17484 LSpanclspn 20969 LSAtomsclsa 38975 Atomscatm 39264 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 TEndoctendo 40754 DVecHcdvh 41080 DIsoCcdic 41174 DIsoHcdih 41230 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tendo 40757 df-edring 40759 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 |
| This theorem is referenced by: (None) |
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