cdlemn7 - Mathbox for Norm Megill

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Theorem cdlemn7 41958

Description: Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

**Hypotheses**
Ref	Expression
cdlemn8.b	⊢ 𝐵 = (Base‘𝐾)
cdlemn8.l	⊢ ≤ = (le‘𝐾)
cdlemn8.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemn8.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemn8.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
cdlemn8.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
cdlemn8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn8.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
cdlemn8.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
cdlemn8.s	⊢ + = (+_g‘𝑈)
cdlemn8.f	⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
cdlemn8.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)

**Assertion**
Ref	Expression
cdlemn7	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝐺 = ((𝑠‘𝐹) ∘ 𝑔) ∧ ( I ↾ 𝑇) = 𝑠))

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

**Proof of Theorem cdlemn7**
Step	Hyp	Ref	Expression
1		simp33 1230	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))
2		simp1 1154	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 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 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 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 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 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 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 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 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑔 ∈ 𝑇)
7		cdlemn8.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
8		cdlemn8.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
9		cdlemn8.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
10		cdlemn8.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
11		cdlemn8.p	. . . . 5 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
12		cdlemn8.o	. . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
13		cdlemn8.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
14		cdlemn8.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
15		cdlemn8.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
16		cdlemn8.s	. . . . 5 ⊢ + = (+_g‘𝑈)
17		cdlemn8.f	. . . . 5 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
18	7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17	cdlemn6 41957	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐹) ∘ 𝑔), 𝑠⟩)
19	2, 3, 4, 5, 6, 18	syl122anc 1406	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐹) ∘ 𝑔), 𝑠⟩)
20	1, 19	eqtrd 2798	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ⟨𝐺, ( I ↾ 𝑇)⟩ = ⟨((𝑠‘𝐹) ∘ 𝑔), 𝑠⟩)
21		fvex 6896	. . . 4 ⊢ (𝑠‘𝐹) ∈ V
22		vex 3459	. . . 4 ⊢ 𝑔 ∈ V
23	21, 22	coex 7928	. . 3 ⊢ ((𝑠‘𝐹) ∘ 𝑔) ∈ V
24		vex 3459	. . 3 ⊢ 𝑠 ∈ V
25	23, 24	opth2 5464	. 2 ⊢ (⟨𝐺, ( I ↾ 𝑇)⟩ = ⟨((𝑠‘𝐹) ∘ 𝑔), 𝑠⟩ ↔ (𝐺 = ((𝑠‘𝐹) ∘ 𝑔) ∧ ( I ↾ 𝑇) = 𝑠))
26	20, 25	sylib 221	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝐺 = ((𝑠‘𝐹) ∘ 𝑔) ∧ ( I ↾ 𝑇) = 𝑠))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ⟨cop 4596 class class class wbr 5110 ↦ cmpt 5193 I cid 5557 ↾ cres 5665 ∘ ccom 5667 ‘cfv 6538 ℩crio 7368 (class class class)co 7412 Basecbs 17270 +_gcplusg 17311 lecple 17318 occoc 17319 Atomscatm 40018 HLchlt 40105 LHypclh 40739 LTrncltrn 40856 TEndoctendo 41507 DVecHcdvh 41833

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-riotaBAD 39708

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 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 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-undef 8270 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-n0 12506 df-z 12593 df-uz 12864 df-fz 13537 df-struct 17208 df-slot 17243 df-ndx 17255 df-base 17271 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-proset 18351 df-poset 18370 df-plt 18385 df-lub 18401 df-glb 18402 df-join 18403 df-meet 18404 df-p0 18480 df-p1 18481 df-lat 18489 df-clat 18556 df-oposet 39931 df-ol 39933 df-oml 39934 df-covers 40021 df-ats 40022 df-atl 40053 df-cvlat 40077 df-hlat 40106 df-llines 40253 df-lplanes 40254 df-lvols 40255 df-lines 40256 df-psubsp 40258 df-pmap 40259 df-padd 40551 df-lhyp 40743 df-laut 40744 df-ldil 40859 df-ltrn 40860 df-trl 40914 df-tendo 41510 df-edring 41512 df-dvech 41834

This theorem is referenced by: cdlemn8 41959

W3C validator