cdlemkfid2N - Mathbox for Norm Megill

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Theorem cdlemkfid2N 42183

Description: Lemma for cdlemkfid3N 42185. (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 1217	. . . . . 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 6913	. . . . 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 7461	. . . 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 7462	. . 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 42181	. . . 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 1196	. . 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 2800	. 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 2810	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 class class class wbr 5106 I cid 5550 ^◡ccnv 5655 ↾ cres 5658 ∘ ccom 5660 ‘cfv 6561 (class class class)co 7446 Basecbs 17419 lecple 17467 joincjn 18517 meetcmee 18518 Atomscatm 40523 HLchlt 40610 LHypclh 41244 LTrncltrn 41361 trLctrl 41418

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-riotaBAD 40213

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-1st 8028 df-2nd 8029 df-undef 8321 df-map 8880 df-proset 18500 df-poset 18519 df-plt 18534 df-lub 18550 df-glb 18551 df-join 18552 df-meet 18553 df-p0 18629 df-p1 18630 df-lat 18638 df-clat 18705 df-oposet 40436 df-ol 40438 df-oml 40439 df-covers 40526 df-ats 40527 df-atl 40558 df-cvlat 40582 df-hlat 40611 df-llines 40758 df-lplanes 40759 df-lvols 40760 df-lines 40761 df-psubsp 40763 df-pmap 40764 df-padd 41056 df-lhyp 41248 df-laut 41249 df-ldil 41364 df-ltrn 41365 df-trl 41419

This theorem is used by: cdlemkfid3N 42185

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