cdlemg17j - Mathbox for Norm Megill

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Theorem cdlemg17j 40144

Description: TODO: fix comment. (Contributed by NM, 11-May-2013.)

**Hypotheses**
Ref	Expression
cdlemg12.l	⊢ ≤ = (le‘𝐾)
cdlemg12.j	⊢ ∨ = (join‘𝐾)
cdlemg12.m	⊢ ∧ = (meet‘𝐾)
cdlemg12.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg12.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg12.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg12b.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemg17j	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘𝑃)) = (𝐹‘(𝐺‘𝑃)))

Distinct variable groups: 𝐴,𝑟 𝐺,𝑟 ∨ ,𝑟 ≤ ,𝑟 𝑃,𝑟 𝑄,𝑟 𝑊,𝑟 𝐹,𝑟 Allowed substitution hints: 𝑅(𝑟) 𝑇(𝑟) 𝐻(𝑟) 𝐾(𝑟) ∧ (𝑟)

**Proof of Theorem cdlemg17j**
Step	Hyp	Ref	Expression
1		cdlemg12.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		cdlemg12.j	. . 3 ⊢ ∨ = (join‘𝐾)
3		cdlemg12.m	. . 3 ⊢ ∧ = (meet‘𝐾)
4		cdlemg12.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
5		cdlemg12.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdlemg12.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7		cdlemg12b.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	cdlemg17i 40142	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘𝑃)) = (𝐹‘𝑄))
9	1, 2, 3, 4, 5, 6, 7	cdlemg17ir 40143	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐹‘(𝐺‘𝑃)) = (𝐹‘𝑄))
10	8, 9	eqtr4d 2771	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘𝑃)) = (𝐹‘(𝐺‘𝑃)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∃wrex 3067 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 lecple 17240 joincjn 18303 meetcmee 18304 Atomscatm 38735 HLchlt 38822 LHypclh 39457 LTrncltrn 39574 trLctrl 39631

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-riotaBAD 38425

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-undef 8279 df-map 8847 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632

This theorem is referenced by: cdlemg17jq 40149 cdlemg17 40150 cdlemg21 40159

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