dihpN - Mathbox for Norm Megill

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Theorem dihpN 35426

Description: The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector ⟨0, 𝑆⟩ (the zero translation ltrnid 34222 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 2609	. 2 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
2		dihp.n	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
3		eqid 2609	. 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 35199	. 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 35425	. 2 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈))
11		eqid 2609	. . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾)
12		eqid 2609	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
13	11, 12, 4, 8	lhpocnel2 34106	. . . . . . . 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 2609	. . . . . . . 8 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)
18	15, 11, 12, 4, 16, 17	ltrniotaidvalN 34672	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵))
19	6, 14, 18	syl2anc 690	. . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵))
20	19	fveq2d 6091	. . . . 5 ⊢ (𝜑 → (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘( I ↾ 𝐵)))
21		dihp.s	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂))
22	21	simpld 473	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐸)
23		dihp.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
24	15, 4, 23	tendoid 34862	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
25	6, 22, 24	syl2anc 690	. . . . 5 ⊢ (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
26	20, 25	eqtr2d 2644	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))
27		fvex 6097	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
28	15, 27	eqeltri 2683	. . . . . 6 ⊢ 𝐵 ∈ V
29		resiexg 6971	. . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
30	28, 29	mp1i 13	. . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V)
31		eqeq1 2613	. . . . . . 7 ⊢ (𝑔 = ( I ↾ 𝐵) → (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))))
32	31	anbi1d 736	. . . . . 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 6086	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))
34	33	eqeq2d 2619	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))))
35		eleq1 2675	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸))
36	34, 35	anbi12d 742	. . . . . 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 4907	. . . . 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 690	. . . 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 958	. . 3 ⊢ (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)})
40		eqid 2609	. . . . . 6 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
41	11, 12, 4, 40, 9	dihvalcqat 35329	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
42	6, 14, 41	syl2anc 690	. . . 4 ⊢ (𝜑 → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
43	11, 12, 4, 8, 16, 23, 40	dicval 35266	. . . . 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 690	. . . 4 ⊢ (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)})
45	42, 44	eqtr2d 2644	. . 3 ⊢ (𝜑 → {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} = (𝐼‘𝑃))
46	39, 45	eleqtrd 2689	. 2 ⊢ (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ∈ (𝐼‘𝑃))
47	21	simprd 477	. . 3 ⊢ (𝜑 → 𝑆 ≠ 𝑂)
48		dihp.o	. . . . . . . 8 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
49	15, 4, 16, 5, 1, 48	dvh0g 35201	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝑈) = ⟨( I ↾ 𝐵), 𝑂⟩)
50	6, 49	syl 17	. . . . . 6 ⊢ (𝜑 → (0_g‘𝑈) = ⟨( I ↾ 𝐵), 𝑂⟩)
51	50	eqeq2d 2619	. . . . 5 ⊢ (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ = (0_g‘𝑈) ↔ ⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩))
52	28, 29	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ 𝐵) ∈ V
53		fvex 6097	. . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
54	16, 53	eqeltri 2683	. . . . . . . . 9 ⊢ 𝑇 ∈ V
55	54	mptex 6367	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
56	48, 55	eqeltri 2683	. . . . . . 7 ⊢ 𝑂 ∈ V
57	52, 56	opth2 4868	. . . . . 6 ⊢ (⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ 𝑆 = 𝑂))
58	57	simprbi 478	. . . . 5 ⊢ (⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ → 𝑆 = 𝑂)
59	51, 58	syl6bi 241	. . . 4 ⊢ (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ = (0_g‘𝑈) → 𝑆 = 𝑂))
60	59	necon3d 2802	. . 3 ⊢ (𝜑 → (𝑆 ≠ 𝑂 → ⟨( I ↾ 𝐵), 𝑆⟩ ≠ (0_g‘𝑈)))
61	47, 60	mpd 15	. 2 ⊢ (𝜑 → ⟨( I ↾ 𝐵), 𝑆⟩ ≠ (0_g‘𝑈))
62	1, 2, 3, 7, 10, 46, 61	lsatel 33093	1 ⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{⟨( I ↾ 𝐵), 𝑆⟩}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 194 ∧ wa 382 = wceq 1474 ∈ wcel 1976 ≠ wne 2779 Vcvv 3172 {csn 4124 ⟨cop 4130 class class class wbr 4577 {copab 4636 ↦ cmpt 4637 I cid 4937 ↾ cres 5029 ‘cfv 5789 ℩crio 6487 Basecbs 15643 lecple 15723 occoc 15724 0_gc0g 15871 LSpanclspn 18740 LSAtomsclsa 33062 Atomscatm 33351 HLchlt 33438 LHypclh 34071 LTrncltrn 34188 TEndoctendo 34841 DVecHcdvh 35168 DIsoCcdic 35262 DIsoHcdih 35318

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2033 ax-13 2233 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4711 ax-pow 4763 ax-pr 4827 ax-un 6824 ax-cnex 9848 ax-resscn 9849 ax-1cn 9850 ax-icn 9851 ax-addcl 9852 ax-addrcl 9853 ax-mulcl 9854 ax-mulrcl 9855 ax-mulcom 9856 ax-addass 9857 ax-mulass 9858 ax-distr 9859 ax-i2m1 9860 ax-1ne0 9861 ax-1rid 9862 ax-rnegex 9863 ax-rrecex 9864 ax-cnre 9865 ax-pre-lttri 9866 ax-pre-lttrn 9867 ax-pre-ltadd 9868 ax-pre-mulgt0 9869 ax-riotaBAD 33040

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-pss 3555 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-tp 4129 df-op 4131 df-uni 4367 df-int 4405 df-iun 4451 df-iin 4452 df-br 4578 df-opab 4638 df-mpt 4639 df-tr 4675 df-eprel 4938 df-id 4942 df-po 4948 df-so 4949 df-fr 4986 df-we 4988 df-xp 5033 df-rel 5034 df-cnv 5035 df-co 5036 df-dm 5037 df-rn 5038 df-res 5039 df-ima 5040 df-pred 5582 df-ord 5628 df-on 5629 df-lim 5630 df-suc 5631 df-iota 5753 df-fun 5791 df-fn 5792 df-f 5793 df-f1 5794 df-fo 5795 df-f1o 5796 df-fv 5797 df-riota 6488 df-ov 6529 df-oprab 6530 df-mpt2 6531 df-om 6935 df-1st 7036 df-2nd 7037 df-tpos 7216 df-undef 7263 df-wrecs 7271 df-recs 7332 df-rdg 7370 df-1o 7424 df-oadd 7428 df-er 7606 df-map 7723 df-en 7819 df-dom 7820 df-sdom 7821 df-fin 7822 df-pnf 9932 df-mnf 9933 df-xr 9934 df-ltxr 9935 df-le 9936 df-sub 10119 df-neg 10120 df-nn 10870 df-2 10928 df-3 10929 df-4 10930 df-5 10931 df-6 10932 df-n0 11142 df-z 11213 df-uz 11522 df-fz 12155 df-struct 15645 df-ndx 15646 df-slot 15647 df-base 15648 df-sets 15649 df-ress 15650 df-plusg 15729 df-mulr 15730 df-sca 15732 df-vsca 15733 df-0g 15873 df-preset 16699 df-poset 16717 df-plt 16729 df-lub 16745 df-glb 16746 df-join 16747 df-meet 16748 df-p0 16810 df-p1 16811 df-lat 16817 df-clat 16879 df-mgm 17013 df-sgrp 17055 df-mnd 17066 df-submnd 17107 df-grp 17196 df-minusg 17197 df-sbg 17198 df-subg 17362 df-cntz 17521 df-lsm 17822 df-cmn 17966 df-abl 17967 df-mgp 18261 df-ur 18273 df-ring 18320 df-oppr 18394 df-dvdsr 18412 df-unit 18413 df-invr 18443 df-dvr 18454 df-drng 18520 df-lmod 18636 df-lss 18702 df-lsp 18741 df-lvec 18872 df-lsatoms 33064 df-oposet 33264 df-ol 33266 df-oml 33267 df-covers 33354 df-ats 33355 df-atl 33386 df-cvlat 33410 df-hlat 33439 df-llines 33585 df-lplanes 33586 df-lvols 33587 df-lines 33588 df-psubsp 33590 df-pmap 33591 df-padd 33883 df-lhyp 34075 df-laut 34076 df-ldil 34191 df-ltrn 34192 df-trl 34247 df-tendo 34844 df-edring 34846 df-disoa 35119 df-dvech 35169 df-dib 35229 df-dic 35263 df-dih 35319

This theorem is referenced by: (None)

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