dihordlem7 - Mathbox for Norm Megill

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Theorem dihordlem7 42463

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 1230	. . 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 1154	. . . 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 1218	. . . 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 1219	. . . 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 1228	. . . 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 1229	. . . 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 42462	. . . 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 1407	. . 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 2798	. 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 6921	. . . 4 ⊢ (𝑠‘𝐺) ∈ V
22		vex 3458	. . . 4 ⊢ 𝑔 ∈ V
23	21, 22	coex 7966	. . 3 ⊢ ((𝑠‘𝐺) ∘ 𝑔) ∈ V
24		vex 3458	. . 3 ⊢ 𝑠 ∈ V
25	23, 24	opth2 5453	. 2 ⊢ (⟨𝑓, 𝑂⟩ = ⟨((𝑠‘𝐺) ∘ 𝑔), 𝑠⟩ ↔ (𝑓 = ((𝑠‘𝐺) ∘ 𝑔) ∧ 𝑂 = 𝑠))
26	20, 25	sylib 221	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ⟨cop 4838 class class class wbr 5106 ↦ cmpt 5189 I cid 5549 ↾ cres 5657 ∘ ccom 5659 ‘cfv 6558 ℩crio 7399 (class class class)co 7443 Basecbs 17416 +_gcplusg 17457 lecple 17464 occoc 17465 Atomscatm 40512 HLchlt 40599 LHypclh 41233 LTrncltrn 41350 TEndoctendo 42001 DVecHcdvh 42327

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-riotaBAD 40202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-undef 8318 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-n0 12634 df-z 12721 df-uz 12993 df-fz 13667 df-struct 17354 df-slot 17389 df-ndx 17401 df-base 17417 df-plusg 17470 df-mulr 17471 df-sca 17473 df-vsca 17474 df-proset 18497 df-poset 18516 df-plt 18531 df-lub 18547 df-glb 18548 df-join 18549 df-meet 18550 df-p0 18626 df-p1 18627 df-lat 18635 df-clat 18702 df-oposet 40425 df-ol 40427 df-oml 40428 df-covers 40515 df-ats 40516 df-atl 40547 df-cvlat 40571 df-hlat 40600 df-llines 40747 df-lplanes 40748 df-lvols 40749 df-lines 40750 df-psubsp 40752 df-pmap 40753 df-padd 41045 df-lhyp 41237 df-laut 41238 df-ldil 41353 df-ltrn 41354 df-trl 41408 df-tendo 42004 df-edring 42006 df-dvech 42328

This theorem is used by: dihordlem7b 42464

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