dihordlem7 - Mathbox for Norm Megill

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Theorem dihordlem7 40579

Description: Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

**Hypotheses**
Ref	Expression
dihordlem8.b	⊢ 𝐵 = (Base‘𝐾)
dihordlem8.l	⊢ ≤ = (le‘𝐾)
dihordlem8.a	⊢ 𝐴 = (Atoms‘𝐾)
dihordlem8.h	⊢ 𝐻 = (LHyp‘𝐾)
dihordlem8.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dihordlem8.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
dihordlem8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihordlem8.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihordlem8.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihordlem8.s	⊢ + = (+_g‘𝑈)
dihordlem8.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)

**Assertion**
Ref	Expression
dihordlem7	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑓 = ((𝑠‘𝐺) ∘ 𝑔) ∧ 𝑂 = 𝑠))

Distinct variable groups: ≤ ,ℎ 𝐴,ℎ 𝐵,ℎ ℎ,𝐻 ℎ,𝐾 𝑃,ℎ 𝑅,ℎ 𝑇,ℎ ℎ,𝑊 Allowed substitution hints: 𝐴(𝑓,𝑔,𝑠) 𝐵(𝑓,𝑔,𝑠) 𝑃(𝑓,𝑔,𝑠) + (𝑓,𝑔,ℎ,𝑠) 𝑄(𝑓,𝑔,ℎ,𝑠) 𝑅(𝑓,𝑔,𝑠) 𝑇(𝑓,𝑔,𝑠) 𝑈(𝑓,𝑔,ℎ,𝑠) 𝐸(𝑓,𝑔,ℎ,𝑠) 𝐺(𝑓,𝑔,ℎ,𝑠) 𝐻(𝑓,𝑔,𝑠) 𝐾(𝑓,𝑔,𝑠) ≤ (𝑓,𝑔,𝑠) 𝑂(𝑓,𝑔,ℎ,𝑠) 𝑊(𝑓,𝑔,𝑠)

**Proof of Theorem dihordlem7**
Step	Hyp	Ref	Expression
1		simp33 1208	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))
2		simp1 1133	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simp2l 1196	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
4		simp2r 1197	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
5		simp31 1206	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑠 ∈ 𝐸)
6		simp32 1207	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑔 ∈ 𝑇)
7		dihordlem8.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
8		dihordlem8.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
9		dihordlem8.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
10		dihordlem8.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
11		dihordlem8.p	. . . . 5 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
12		dihordlem8.o	. . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
13		dihordlem8.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
14		dihordlem8.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
15		dihordlem8.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
16		dihordlem8.s	. . . . 5 ⊢ + = (+_g‘𝑈)
17		dihordlem8.g	. . . . 5 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)
18	7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17	dihordlem6 40578	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐺) ∘ 𝑔), 𝑠⟩)
19	2, 3, 4, 5, 6, 18	syl122anc 1376	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐺) ∘ 𝑔), 𝑠⟩)
20	1, 19	eqtrd 2764	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ⟨𝑓, 𝑂⟩ = ⟨((𝑠‘𝐺) ∘ 𝑔), 𝑠⟩)
21		fvex 6895	. . . 4 ⊢ (𝑠‘𝐺) ∈ V
22		vex 3470	. . . 4 ⊢ 𝑔 ∈ V
23	21, 22	coex 7915	. . 3 ⊢ ((𝑠‘𝐺) ∘ 𝑔) ∈ V
24		vex 3470	. . 3 ⊢ 𝑠 ∈ V
25	23, 24	opth2 5471	. 2 ⊢ (⟨𝑓, 𝑂⟩ = ⟨((𝑠‘𝐺) ∘ 𝑔), 𝑠⟩ ↔ (𝑓 = ((𝑠‘𝐺) ∘ 𝑔) ∧ 𝑂 = 𝑠))
26	20, 25	sylib 217	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑓 = ((𝑠‘𝐺) ∘ 𝑔) ∧ 𝑂 = 𝑠))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⟨cop 4627 class class class wbr 5139 ↦ cmpt 5222 I cid 5564 ↾ cres 5669 ∘ ccom 5671 ‘cfv 6534 ℩crio 7357 (class class class)co 7402 Basecbs 17145 +_gcplusg 17198 lecple 17205 occoc 17206 Atomscatm 38627 HLchlt 38714 LHypclh 39349 LTrncltrn 39466 TEndoctendo 40117 DVecHcdvh 40443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38317

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-struct 17081 df-slot 17116 df-ndx 17128 df-base 17146 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38540 df-ol 38542 df-oml 38543 df-covers 38630 df-ats 38631 df-atl 38662 df-cvlat 38686 df-hlat 38715 df-llines 38863 df-lplanes 38864 df-lvols 38865 df-lines 38866 df-psubsp 38868 df-pmap 38869 df-padd 39161 df-lhyp 39353 df-laut 39354 df-ldil 39469 df-ltrn 39470 df-trl 39524 df-tendo 40120 df-edring 40122 df-dvech 40444

This theorem is referenced by: dihordlem7b 40580

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