cdlemkfid2N - Mathbox for Norm Megill

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Theorem cdlemkfid2N 40882

Description: Lemma for cdlemkfid3N 40884. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

**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))))

**Assertion**
Ref	Expression
cdlemkfid2N	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 = 𝑁) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏 ∈ 𝑇) ∧ ((𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝑍 = (𝑏‘𝑃))

**Proof of Theorem cdlemkfid2N**
Step	Hyp	Ref	Expression
1		cdlemk5.z	. 2 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
2		simp1r 1198	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 = 𝑁) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏 ∈ 𝑇) ∧ ((𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐹 = 𝑁)
3	2	fveq1d 6924	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 = 𝑁) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏 ∈ 𝑇) ∧ ((𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝐹‘𝑃) = (𝑁‘𝑃))
4	3	oveq1d 7465	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 = 𝑁) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏 ∈ 𝑇) ∧ ((𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐹‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))) = ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
5	4	oveq2d 7466	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 = 𝑁) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏 ∈ 𝑇) ∧ ((𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝐹‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹)))) = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹)))))
6		cdlemk5.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
7		cdlemk5.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
8		cdlemk5.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
9		cdlemk5.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
10		cdlemk5.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
11		cdlemk5.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
12		cdlemk5.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13		cdlemk5.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
14	6, 7, 8, 9, 10, 11, 12, 13	cdlemkfid1N 40880	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏 ∈ 𝑇) ∧ ((𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝐹‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹)))) = (𝑏‘𝑃))
15	14	3adant1r 1177	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 = 𝑁) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏 ∈ 𝑇) ∧ ((𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝐹‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹)))) = (𝑏‘𝑃))
16	5, 15	eqtr3d 2782	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 = 𝑁) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏 ∈ 𝑇) ∧ ((𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹)))) = (𝑏‘𝑃))
17	1, 16	eqtrid 2792	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 = 𝑁) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏 ∈ 𝑇) ∧ ((𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝑍 = (𝑏‘𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 I cid 5592 ^◡ccnv 5699 ↾ cres 5702 ∘ ccom 5704 ‘cfv 6575 (class class class)co 7450 Basecbs 17260 lecple 17320 joincjn 18383 meetcmee 18384 Atomscatm 39221 HLchlt 39308 LHypclh 39943 LTrncltrn 40060 trLctrl 40117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-riotaBAD 38911

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1st 8032 df-2nd 8033 df-undef 8316 df-map 8888 df-proset 18367 df-poset 18385 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18504 df-clat 18571 df-oposet 39134 df-ol 39136 df-oml 39137 df-covers 39224 df-ats 39225 df-atl 39256 df-cvlat 39280 df-hlat 39309 df-llines 39457 df-lplanes 39458 df-lvols 39459 df-lines 39460 df-psubsp 39462 df-pmap 39463 df-padd 39755 df-lhyp 39947 df-laut 39948 df-ldil 40063 df-ltrn 40064 df-trl 40118

This theorem is referenced by: cdlemkfid3N 40884

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