dihpN - Mathbox for Norm Megill

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Theorem dihpN 39277

Description: The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector ⟨0, 𝑆⟩ (the zero translation ltrnid 38076 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.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
dihpN	⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{⟨( I ↾ 𝐵), 𝑆⟩}))

Distinct variable groups: 𝐵,𝑓 𝑓,𝐻 𝑓,𝐾 𝑃,𝑓 𝑇,𝑓 𝑓,𝑊 Allowed substitution hints: 𝜑(𝑓) 𝑆(𝑓) 𝑈(𝑓) 𝐸(𝑓) 𝐼(𝑓) 𝑁(𝑓) 𝑂(𝑓)

Proof of Theorem dihpN Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. 2 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
2		dihp.n	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
3		eqid 2738	. 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 39050	. 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 39276	. 2 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈))
11		eqid 2738	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
12		eqid 2738	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
13	11, 12, 4, 8	lhpocnel2 37960	. . . . . . 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 2738	. . . . . . . 8 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)
17	14, 11, 12, 4, 15, 16	ltrniotaidvalN 38524	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵))
18	6, 13, 17	syl2anc2 584	. . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵))
19	18	fveq2d 6760	. . . . 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 38714	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
24	6, 21, 23	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
25	19, 24	eqtr2d 2779	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))
26	14	fvexi 6770	. . . . . 6 ⊢ 𝐵 ∈ V
27		resiexg 7735	. . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
28	26, 27	mp1i 13	. . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V)
29		eqeq1 2742	. . . . . . 7 ⊢ (𝑔 = ( I ↾ 𝐵) → (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))))
30	29	anbi1d 629	. . . . . 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 6755	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))
32	31	eqeq2d 2749	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))))
33		eleq1 2826	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸))
34	32, 33	anbi12d 630	. . . . . 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 5444	. . . . 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 583	. . . 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 709	. . 3 ⊢ (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)})
38		eqid 2738	. . . . . 6 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
39	11, 12, 4, 38, 9	dihvalcqat 39180	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
40	6, 13, 39	syl2anc2 584	. . . 4 ⊢ (𝜑 → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
41	11, 12, 4, 8, 15, 22, 38	dicval 39117	. . . . 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 584	. . . 4 ⊢ (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)})
43	40, 42	eqtr2d 2779	. . 3 ⊢ (𝜑 → {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} = (𝐼‘𝑃))
44	37, 43	eleqtrd 2841	. 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 39052	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝑈) = ⟨( I ↾ 𝐵), 𝑂⟩)
48	6, 47	syl 17	. . . . . 6 ⊢ (𝜑 → (0_g‘𝑈) = ⟨( I ↾ 𝐵), 𝑂⟩)
49	48	eqeq2d 2749	. . . . 5 ⊢ (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ = (0_g‘𝑈) ↔ ⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩))
50	26, 27	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ 𝐵) ∈ V
51	15	fvexi 6770	. . . . . . . . 9 ⊢ 𝑇 ∈ V
52	51	mptex 7081	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
53	46, 52	eqeltri 2835	. . . . . . 7 ⊢ 𝑂 ∈ V
54	50, 53	opth2 5389	. . . . . 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	syl6bi 252	. . . 4 ⊢ (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ = (0_g‘𝑈) → 𝑆 = 𝑂))
57	56	necon3d 2963	. . 3 ⊢ (𝜑 → (𝑆 ≠ 𝑂 → ⟨( I ↾ 𝐵), 𝑆⟩ ≠ (0_g‘𝑈)))
58	45, 57	mpd 15	. 2 ⊢ (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ≠ (0_g‘𝑈))
59	1, 2, 3, 7, 10, 44, 58	lsatel 36946	1 ⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{⟨( I ↾ 𝐵), 𝑆⟩}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 Vcvv 3422 {csn 4558 ⟨cop 4564 class class class wbr 5070 {copab 5132 ↦ cmpt 5153 I cid 5479 ↾ cres 5582 ‘cfv 6418 ℩crio 7211 Basecbs 16840 lecple 16895 occoc 16896 0_gc0g 17067 LSpanclspn 20148 LSAtomsclsa 36915 Atomscatm 37204 HLchlt 37291 LHypclh 37925 LTrncltrn 38042 TEndoctendo 38693 DVecHcdvh 39019 DIsoCcdic 39113 DIsoHcdih 39169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tendo 38696 df-edring 38698 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170

This theorem is referenced by: (None)

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