cdlemk19u - Mathbox for Norm Megill

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Theorem cdlemk19u 40145

Description: Part of Lemma K of [Crawley] p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with 𝐹, 𝑁, 𝑈. (Contributed by NM, 31-Jul-2013.)

**Hypotheses**
Ref	Expression
cdlemk5.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk5.l	⊢ ≤ = (le‘𝐾)
cdlemk5.j	⊢ ∨ = (join‘𝐾)
cdlemk5.m	⊢ ∧ = (meet‘𝐾)
cdlemk5.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk5.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk5.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk5.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk5.z	⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
cdlemk5.y	⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
cdlemk5.x	⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌))
cdlemk5.u	⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))

**Assertion**
Ref	Expression
cdlemk19u	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑈‘𝐹) = 𝑁)

Distinct variable groups: ∧ ,𝑔 ∨ ,𝑔 𝐵,𝑔 𝑃,𝑔 𝑅,𝑔 𝑇,𝑔 𝑔,𝑍 𝑔,𝑏,𝑧, ∧ ≤ ,𝑏 𝑧,𝑔, ≤ ∨ ,𝑏,𝑧 𝐴,𝑏,𝑔,𝑧 𝐵,𝑏,𝑧 𝐹,𝑏,𝑔,𝑧 𝐻,𝑏,𝑔,𝑧 𝐾,𝑏,𝑔,𝑧 𝑁,𝑏,𝑔,𝑧 𝑃,𝑏,𝑧 𝑅,𝑏,𝑧 𝑇,𝑏,𝑧 𝑊,𝑏,𝑔,𝑧 𝑧,𝑌 Allowed substitution hints: 𝑈(𝑧,𝑔,𝑏) 𝑋(𝑧,𝑔,𝑏) 𝑌(𝑔,𝑏) 𝑍(𝑧,𝑏)

**Proof of Theorem cdlemk19u**
Step	Hyp	Ref	Expression
1		simp1l 1196	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp1 1135	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)))
3		simp2l 1198	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
4		simp2r 1199	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑁 ∈ 𝑇)
5		simp3 1137	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
6		cdlemk5.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
7		cdlemk5.l	. . . 4 ⊢ ≤ = (le‘𝐾)
8		cdlemk5.j	. . . 4 ⊢ ∨ = (join‘𝐾)
9		cdlemk5.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
10		cdlemk5.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
11		cdlemk5.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
12		cdlemk5.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13		cdlemk5.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
14		cdlemk5.z	. . . 4 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
15		cdlemk5.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
16		cdlemk5.x	. . . 4 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌))
17		cdlemk5.u	. . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
18	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17	cdlemk35u 40139	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑈‘𝐹) ∈ 𝑇)
19	2, 3, 4, 3, 5, 18	syl131anc 1382	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑈‘𝐹) ∈ 𝑇)
20		simpr 484	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → 𝐹 = 𝑁)
21		simpl2l 1225	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → 𝐹 ∈ 𝑇)
22	16, 17	cdlemk40t 40093	. . . . . 6 ⊢ ((𝐹 = 𝑁 ∧ 𝐹 ∈ 𝑇) → (𝑈‘𝐹) = 𝐹)
23	20, 21, 22	syl2anc 583	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → (𝑈‘𝐹) = 𝐹)
24	23	fveq1d 6893	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → ((𝑈‘𝐹)‘𝑃) = (𝐹‘𝑃))
25		fveq1 6890	. . . . 5 ⊢ (𝐹 = 𝑁 → (𝐹‘𝑃) = (𝑁‘𝑃))
26	25	adantl 481	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → (𝐹‘𝑃) = (𝑁‘𝑃))
27	24, 26	eqtrd 2771	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 = 𝑁) → ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃))
28		simpl1 1190	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)))
29		simpl2l 1225	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → 𝐹 ∈ 𝑇)
30		simpr 484	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → 𝐹 ≠ 𝑁)
31		simpl2r 1226	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → 𝑁 ∈ 𝑇)
32		simpl3 1192	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
33	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17	cdlemk19u1 40144	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ 𝑁 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃))
34	28, 29, 30, 31, 32, 33	syl131anc 1382	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ≠ 𝑁) → ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃))
35	27, 34	pm2.61dane 3028	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃))
36	7, 10, 11, 12	cdlemd 39382	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈‘𝐹) ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ ((𝑈‘𝐹)‘𝑃) = (𝑁‘𝑃)) → (𝑈‘𝐹) = 𝑁)
37	1, 19, 4, 5, 35, 36	syl311anc 1383	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑈‘𝐹) = 𝑁)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 ^◡ccnv 5675 ↾ cres 5678 ∘ ccom 5680 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 Basecbs 17149 lecple 17209 joincjn 18269 meetcmee 18270 Atomscatm 38437 HLchlt 38524 LHypclh 39159 LTrncltrn 39276 trLctrl 39333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-riotaBAD 38127

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-undef 8262 df-map 8826 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 df-laut 39164 df-ldil 39279 df-ltrn 39280 df-trl 39334

This theorem is referenced by: cdlemk19w 40147

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