dihpN - Mathbox for Norm Megill

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Theorem dihpN 41319

Description: The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector ⟨0, 𝑆⟩ (the zero translation ltrnid 40118 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 2735	. 2 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
2		dihp.n	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
3		eqid 2735	. 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 41092	. 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 41318	. 2 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSAtoms‘𝑈))
11		eqid 2735	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
12		eqid 2735	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
13	11, 12, 4, 8	lhpocnel2 40002	. . . . . . 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 2735	. . . . . . . 8 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)
17	14, 11, 12, 4, 15, 16	ltrniotaidvalN 40566	. . . . . . 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 6911	. . . . 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 40756	. . . . . 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 2776	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))
26	14	fvexi 6921	. . . . . 6 ⊢ 𝐵 ∈ V
27		resiexg 7935	. . . . . 6 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
28	26, 27	mp1i 13	. . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V)
29		eqeq1 2739	. . . . . . 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 6906	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)))
32	31	eqeq2d 2746	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (( I ↾ 𝐵) = (𝑠‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃)) ↔ ( I ↾ 𝐵) = (𝑆‘(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃))))
33		eleq1 2827	. . . . . . 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 5548	. . . . 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 2735	. . . . . 6 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
39	11, 12, 4, 38, 9	dihvalcqat 41222	. . . . 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 41159	. . . . 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 2776	. . 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 41094	. . . . . . 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 2746	. . . . 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 6921	. . . . . . . . 9 ⊢ 𝑇 ∈ V
52	51	mptex 7243	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
53	46, 52	eqeltri 2835	. . . . . . 7 ⊢ 𝑂 ∈ V
54	50, 53	opth2 5491	. . . . . 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⟩ = (0_g‘𝑈) → 𝑆 = 𝑂))
57	56	necon3d 2959	. . 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 38987	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 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 {csn 4631 ⟨cop 4637 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 I cid 5582 ↾ cres 5691 ‘cfv 6563 ℩crio 7387 Basecbs 17245 lecple 17305 occoc 17306 0_gc0g 17486 LSpanclspn 20987 LSAtomsclsa 38956 Atomscatm 39245 HLchlt 39332 LHypclh 39967 LTrncltrn 40084 TEndoctendo 40735 DVecHcdvh 41061 DIsoCcdic 41155 DIsoHcdih 41211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tendo 40738 df-edring 40740 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212

This theorem is referenced by: (None)

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