cdlemk39u1 - Mathbox for Norm Megill

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Theorem cdlemk39u1 38981

Description: Lemma for cdlemk39u 38982. (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
cdlemk39u1	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑈‘𝐺)) ≤ (𝑅‘𝐺))

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

**Proof of Theorem cdlemk39u1**
Step	Hyp	Ref	Expression
1		simp22 1206	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ≠ 𝑁)
2		simp23 1207	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
3		cdlemk5.x	. . . . 5 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌))
4		cdlemk5.u	. . . . 5 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
5	3, 4	cdlemk40f 38933	. . . 4 ⊢ ((𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌𝑋)
6	1, 2, 5	syl2anc 584	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑈‘𝐺) = ⦋𝐺 / 𝑔⦌𝑋)
7	6	fveq2d 6778	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑈‘𝐺)) = (𝑅‘⦋𝐺 / 𝑔⦌𝑋))
8		simp11 1202	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simp12 1203	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
10		simp13 1204	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑁 ∈ 𝑇)
11		simp21 1205	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = (𝑅‘𝑁))
12		cdlemk5.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
13		cdlemk5.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
14		cdlemk5.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
15		cdlemk5.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
16	12, 13, 14, 15	trlnid 38193	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝐹 ≠ 𝑁 ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → 𝐹 ≠ ( I ↾ 𝐵))
17	8, 9, 10, 1, 11, 16	syl122anc 1378	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ≠ ( I ↾ 𝐵))
18	9, 17	jca 512	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)))
19		simp3 1137	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
20		cdlemk5.l	. . . 4 ⊢ ≤ = (le‘𝐾)
21		cdlemk5.j	. . . 4 ⊢ ∨ = (join‘𝐾)
22		cdlemk5.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
23		cdlemk5.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
24		cdlemk5.z	. . . 4 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
25		cdlemk5.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
26	12, 20, 21, 22, 23, 13, 14, 15, 24, 25, 3	cdlemk39s-id 38954	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝐺 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) → (𝑅‘⦋𝐺 / 𝑔⦌𝑋) ≤ (𝑅‘𝐺))
27	8, 18, 2, 10, 19, 11, 26	syl132anc 1387	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘⦋𝐺 / 𝑔⦌𝑋) ≤ (𝑅‘𝐺))
28	7, 27	eqbrtrd 5096	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐹 ≠ 𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑈‘𝐺)) ≤ (𝑅‘𝐺))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ⦋csb 3832 ifcif 4459 class class class wbr 5074 ↦ cmpt 5157 I cid 5488 ^◡ccnv 5588 ↾ cres 5591 ∘ ccom 5593 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 Basecbs 16912 lecple 16969 joincjn 18029 meetcmee 18030 Atomscatm 37277 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 trLctrl 38172

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-undef 8089 df-map 8617 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173

This theorem is referenced by: cdlemk39u 38982

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