dihpN - Mathbox for Norm Megill

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Theorem dihpN 41716

Description: The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector ⟨0, 𝑆⟩ (the zero translation ltrnid 40515 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.)

**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 2737	. 2 ⊢ (0_g‘𝑈) = (0_g‘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 41489	. 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 41715	. 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 40399	. . . . . . 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 40963	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵))
18	6, 13, 17	syl2anc2 586	. . . . . 6 ⊢ (𝜑 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = ( I ↾ 𝐵))
19	18	fveq2d 6846	. . . . 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 41153	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
24	6, 21, 23	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
25	19, 24	eqtr2d 2773	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))
26	14	fvexi 6856	. . . . . 6 ⊢ 𝐵 ∈ V
27		resiexg 7864	. . . . . 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 632	. . . . . 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 6841	. . . . . . . 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 2825	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸))
34	32, 33	anbi12d 633	. . . . . 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 5494	. . . . 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 585	. . . 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 714	. . 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 41619	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Atoms‘𝐾) ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
40	6, 13, 39	syl2anc2 586	. . . 4 ⊢ (𝜑 → (𝐼‘𝑃) = (((DIsoC‘𝐾)‘𝑊)‘𝑃))
41	11, 12, 4, 8, 15, 22, 38	dicval 41556	. . . . 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 586	. . . 4 ⊢ (𝜑 → (((DIsoC‘𝐾)‘𝑊)‘𝑃) = {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)})
43	40, 42	eqtr2d 2773	. . 3 ⊢ (𝜑 → {⟨𝑔, 𝑠⟩ ∣ (𝑔 = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ∧ 𝑠 ∈ 𝐸)} = (𝐼‘𝑃))
44	37, 43	eleqtrd 2839	. 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 41491	. . . . . . 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 2748	. . . . 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 6856	. . . . . . . . 9 ⊢ 𝑇 ∈ V
52	51	mptex 7179	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
53	46, 52	eqeltri 2833	. . . . . . 7 ⊢ 𝑂 ∈ V
54	50, 53	opth2 5436	. . . . . 6 ⊢ (⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (( I ↾ 𝐵) = ( I ↾ 𝐵) ∧ 𝑆 = 𝑂))
55	54	simprbi 497	. . . . 5 ⊢ (⟨( I ↾ 𝐵), 𝑆⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ → 𝑆 = 𝑂)
56	49, 55	biimtrdi 253	. . . 4 ⊢ (𝜑 → (⟨( I ↾ 𝐵), 𝑆⟩ = (0_g‘𝑈) → 𝑆 = 𝑂))
57	56	necon3d 2954	. . 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 39385	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 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 {csn 4582 ⟨cop 4588 class class class wbr 5100 {copab 5162 ↦ cmpt 5181 I cid 5526 ↾ cres 5634 ‘cfv 6500 ℩crio 7324 Basecbs 17148 lecple 17196 occoc 17197 0_gc0g 17371 LSpanclspn 20937 LSAtomsclsa 39354 Atomscatm 39643 HLchlt 39730 LHypclh 40364 LTrncltrn 40481 TEndoctendo 41132 DVecHcdvh 41458 DIsoCcdic 41552 DIsoHcdih 41608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39333

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-0g 17373 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18881 df-minusg 18882 df-sbg 18883 df-subg 19068 df-cntz 19261 df-lsm 19580 df-cmn 19726 df-abl 19727 df-mgp 20091 df-rng 20103 df-ur 20132 df-ring 20185 df-oppr 20288 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20679 df-lmod 20828 df-lss 20898 df-lsp 20938 df-lvec 21070 df-lsatoms 39356 df-oposet 39556 df-ol 39558 df-oml 39559 df-covers 39646 df-ats 39647 df-atl 39678 df-cvlat 39702 df-hlat 39731 df-llines 39878 df-lplanes 39879 df-lvols 39880 df-lines 39881 df-psubsp 39883 df-pmap 39884 df-padd 40176 df-lhyp 40368 df-laut 40369 df-ldil 40484 df-ltrn 40485 df-trl 40539 df-tendo 41135 df-edring 41137 df-disoa 41409 df-dvech 41459 df-dib 41519 df-dic 41553 df-dih 41609

This theorem is referenced by: (None)

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