cdleme50f1o - Mathbox for Norm Megill

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Theorem cdleme50f1o 36355

Description: Part of proof of Lemma D in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 9-Apr-2013.)

**Hypotheses**
Ref	Expression
cdlemef50.b	⊢ 𝐵 = (Base‘𝐾)
cdlemef50.l	⊢ ≤ = (le‘𝐾)
cdlemef50.j	⊢ ∨ = (join‘𝐾)
cdlemef50.m	⊢ ∧ = (meet‘𝐾)
cdlemef50.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemef50.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemef50.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdlemef50.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdlemefs50.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemef50.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))

**Assertion**
Ref	Expression
cdleme50f1o	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹:𝐵–1-1-onto→𝐵)

Distinct variable groups: 𝑡,𝑠,𝑥,𝑦,𝑧, ∧ ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧 ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝐴,𝑠,𝑡,𝑥,𝑦,𝑧 𝐵,𝑠,𝑡,𝑥,𝑦,𝑧 𝐷,𝑠,𝑥,𝑦,𝑧 𝑥,𝐸,𝑦,𝑧 𝐻,𝑠,𝑡,𝑥,𝑦,𝑧 𝐾,𝑠,𝑡,𝑥,𝑦,𝑧 𝑃,𝑠,𝑡,𝑥,𝑦,𝑧 𝑄,𝑠,𝑡,𝑥,𝑦,𝑧 𝑈,𝑠,𝑡,𝑥,𝑦,𝑧 𝑊,𝑠,𝑡,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐷(𝑡) 𝐸(𝑡,𝑠) 𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)

**Proof of Theorem cdleme50f1o**
Step	Hyp	Ref	Expression
1		cdlemef50.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemef50.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdlemef50.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdlemef50.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdlemef50.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemef50.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemef50.u	. . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdlemef50.d	. . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
9		cdlemefs50.e	. . 3 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
10		cdlemef50.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme50f1 36352	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹:𝐵–1-1→𝐵)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme50rn 36354	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ran 𝐹 = 𝐵)
13		dff1o5 6287	. 2 ⊢ (𝐹:𝐵–1-1-onto→𝐵 ↔ (𝐹:𝐵–1-1→𝐵 ∧ ran 𝐹 = 𝐵))
14	11, 12, 13	sylanbrc 572	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹:𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∀wral 3061 ⦋csb 3682 ifcif 4225 class class class wbr 4786 ↦ cmpt 4863 ran crn 5250 –1-1→wf1 6028 –1-1-onto→wf1o 6030 ‘cfv 6031 ℩crio 6753 (class class class)co 6793 Basecbs 16064 lecple 16156 joincjn 17152 meetcmee 17153 Atomscatm 35072 HLchlt 35159 LHypclh 35792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-riotaBAD 34761

This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-1st 7315 df-2nd 7316 df-undef 7551 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-p1 17248 df-lat 17254 df-clat 17316 df-oposet 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-llines 35306 df-lplanes 35307 df-lvols 35308 df-lines 35309 df-psubsp 35311 df-pmap 35312 df-padd 35604 df-lhyp 35796

This theorem is referenced by: cdleme50laut 36356 cdleme51finvfvN 36364 cdleme51finvN 36365

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