dihordlem7b - Mathbox for Norm Megill

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Theorem dihordlem7b 42348

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
dihordlem7b	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑓 = 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 dihordlem7b**
Step	Hyp	Ref	Expression
1		dihordlem8.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		dihordlem8.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		dihordlem8.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4		dihordlem8.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		dihordlem8.p	. . . . 5 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
6		dihordlem8.o	. . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
7		dihordlem8.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		dihordlem8.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
9		dihordlem8.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
10		dihordlem8.s	. . . . 5 ⊢ + = (+_g‘𝑈)
11		dihordlem8.g	. . . . 5 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	dihordlem7 42347	. . . 4 ⊢ (((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))
13	12	simpld 500	. . 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 = ((𝑠‘𝐺) ∘ 𝑔))
14	12	simprd 501	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑂 = 𝑠)
15	14	fveq1d 6887	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑂‘𝐺) = (𝑠‘𝐺))
16		simp1 1154	. . . . . . 7 ⊢ (((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))
17	2, 3, 4, 5	lhpocnel2 41152	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
18	17	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
19		simp2r 1219	. . . . . . 7 ⊢ (((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))
20	2, 3, 4, 7, 11	ltrniotacl 41712	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
21	16, 18, 19, 20	syl3anc 1398	. . . . . 6 ⊢ (((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)
22	6, 1	tendo02 41920	. . . . . 6 ⊢ (𝐺 ∈ 𝑇 → (𝑂‘𝐺) = ( I ↾ 𝐵))
23	21, 22	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑂‘𝐺) = ( I ↾ 𝐵))
24	15, 23	eqtr3d 2797	. . . 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‘𝐺) = ( I ↾ 𝐵))
25	24	coeq1d 5838	. . 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) ∘ 𝑔) = (( I ↾ 𝐵) ∘ 𝑔))
26		simp32 1229	. . . . 5 ⊢ (((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)
27	1, 4, 7	ltrn1o 41257	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐵–1-1-onto→𝐵)
28	16, 26, 27	syl2anc 596	. . . 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:𝐵–1-1-onto→𝐵)
29		f1of 6824	. . . 4 ⊢ (𝑔:𝐵–1-1-onto→𝐵 → 𝑔:𝐵⟶𝐵)
30		fcoi2 6757	. . . 4 ⊢ (𝑔:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝑔) = 𝑔)
31	28, 29, 30	3syl 19	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (( I ↾ 𝐵) ∘ 𝑔) = 𝑔)
32	13, 25, 31	3eqtrd 2799	. 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 = 𝑔)
33	32, 14	jca 521	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 = 𝑔 ∧ 𝑂 = 𝑠))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ⟨cop 4589 class class class wbr 5102 ↦ cmpt 5185 I cid 5544 ↾ cres 5652 ∘ ccom 5654 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 ℩crio 7376 (class class class)co 7420 Basecbs 17388 +_gcplusg 17429 lecple 17436 occoc 17437 Atomscatm 40396 HLchlt 40483 LHypclh 41117 LTrncltrn 41234 TEndoctendo 41885 DVecHcdvh 42211

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-riotaBAD 40086

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-undef 8291 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-er 8718 df-map 8850 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-n0 12608 df-z 12695 df-uz 12967 df-fz 13641 df-struct 17326 df-slot 17361 df-ndx 17373 df-base 17389 df-plusg 17442 df-mulr 17443 df-sca 17445 df-vsca 17446 df-proset 18469 df-poset 18488 df-plt 18503 df-lub 18519 df-glb 18520 df-join 18521 df-meet 18522 df-p0 18598 df-p1 18599 df-lat 18607 df-clat 18674 df-oposet 40309 df-ol 40311 df-oml 40312 df-covers 40399 df-ats 40400 df-atl 40431 df-cvlat 40455 df-hlat 40484 df-llines 40631 df-lplanes 40632 df-lvols 40633 df-lines 40634 df-psubsp 40636 df-pmap 40637 df-padd 40929 df-lhyp 41121 df-laut 41122 df-ldil 41237 df-ltrn 41238 df-trl 41292 df-tendo 41888 df-edring 41890 df-dvech 42212

This theorem is used by: dihord10 42356

W3C validator