dihordlem7 - Mathbox for Norm Megill

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Theorem dihordlem7 41179

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 1212	. . 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 1136	. . . 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 1200	. . . 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 1201	. . . 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 1210	. . . 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 1211	. . . 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 41178	. . . 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 1381	. . 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 2770	. 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 6888	. . . 4 ⊢ (𝑠‘𝐺) ∈ V
22		vex 3463	. . . 4 ⊢ 𝑔 ∈ V
23	21, 22	coex 7924	. . 3 ⊢ ((𝑠‘𝐺) ∘ 𝑔) ∈ V
24		vex 3463	. . 3 ⊢ 𝑠 ∈ V
25	23, 24	opth2 5455	. 2 ⊢ (⟨𝑓, 𝑂⟩ = ⟨((𝑠‘𝐺) ∘ 𝑔), 𝑠⟩ ↔ (𝑓 = ((𝑠‘𝐺) ∘ 𝑔) ∧ 𝑂 = 𝑠))
26	20, 25	sylib 218	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 1086 = wceq 1540 ∈ wcel 2108 ⟨cop 4607 class class class wbr 5119 ↦ cmpt 5201 I cid 5547 ↾ cres 5656 ∘ ccom 5658 ‘cfv 6530 ℩crio 7359 (class class class)co 7403 Basecbs 17226 +_gcplusg 17269 lecple 17276 occoc 17277 Atomscatm 39227 HLchlt 39314 LHypclh 39949 LTrncltrn 40066 TEndoctendo 40717 DVecHcdvh 41043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-riotaBAD 38917

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-undef 8270 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8717 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-n0 12500 df-z 12587 df-uz 12851 df-fz 13523 df-struct 17164 df-slot 17199 df-ndx 17211 df-base 17227 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-proset 18304 df-poset 18323 df-plt 18338 df-lub 18354 df-glb 18355 df-join 18356 df-meet 18357 df-p0 18433 df-p1 18434 df-lat 18440 df-clat 18507 df-oposet 39140 df-ol 39142 df-oml 39143 df-covers 39230 df-ats 39231 df-atl 39262 df-cvlat 39286 df-hlat 39315 df-llines 39463 df-lplanes 39464 df-lvols 39465 df-lines 39466 df-psubsp 39468 df-pmap 39469 df-padd 39761 df-lhyp 39953 df-laut 39954 df-ldil 40069 df-ltrn 40070 df-trl 40124 df-tendo 40720 df-edring 40722 df-dvech 41044

This theorem is referenced by: dihordlem7b 41180

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