cdlemkyu - Mathbox for Norm Megill

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Theorem cdlemkyu 41297

Description: Convert between function and explicit forms. 𝐶 represents 𝑍 in cdlemkuu 41265. TODO: Clean all this up. (Contributed by NM, 21-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))))
cdlemk5b.s	⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
cdlemk5b.u1	⊢ 𝑉 = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑑))))))
cdlemk5.o2	⊢ 𝑄 = (𝑆‘𝑏)
cdlemk5.u2	⊢ 𝐶 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑏))))))

**Assertion**
Ref	Expression
cdlemkyu	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)))) → ⦋𝐺 / 𝑔⦌𝑌 = ((𝐶‘𝐺)‘𝑃))

Distinct variable groups: ∧ ,𝑔 ∨ ,𝑔 𝐵,𝑔 𝑃,𝑔 𝑅,𝑔 𝑇,𝑔 𝑔,𝑍 𝑔,𝑏 𝑔,𝐺,𝑑,𝑒 𝑓,𝑔,𝑖,𝑗 𝑒,𝑑,𝑓,𝑖,𝑗, ∧ ≤ ,𝑖,𝑗 ∨ ,𝑑,𝑒,𝑓,𝑖,𝑗 𝐴,𝑖,𝑗 𝑓,𝐹,𝑖,𝑗 𝐺,𝑑,𝑒,𝑗 𝑖,𝐻,𝑗 𝑖,𝐾,𝑗 𝑓,𝑁,𝑖,𝑗 𝑃,𝑑,𝑒,𝑓,𝑖,𝑗 𝑅,𝑑,𝑒,𝑓,𝑖,𝑗 𝑏,𝑑,𝑒,𝑗,𝑆 𝑇,𝑑,𝑒,𝑓,𝑖,𝑗 𝑊,𝑑,𝑒,𝑓,𝑖,𝑗 𝑓,𝑏,𝑖 𝑄,𝑑,𝑒 Allowed substitution hints: 𝐴(𝑒,𝑓,𝑔,𝑏,𝑑) 𝐵(𝑒,𝑓,𝑖,𝑗,𝑏,𝑑) 𝐶(𝑒,𝑓,𝑔,𝑖,𝑗,𝑏,𝑑) 𝑃(𝑏) 𝑄(𝑓,𝑔,𝑖,𝑗,𝑏) 𝑅(𝑏) 𝑆(𝑓,𝑔,𝑖) 𝑇(𝑏) 𝐹(𝑒,𝑔,𝑏,𝑑) 𝐺(𝑓,𝑖,𝑏) 𝐻(𝑒,𝑓,𝑔,𝑏,𝑑) ∨ (𝑏) 𝐾(𝑒,𝑓,𝑔,𝑏,𝑑) ≤ (𝑒,𝑓,𝑔,𝑏,𝑑) ∧ (𝑏) 𝑁(𝑒,𝑔,𝑏,𝑑) 𝑉(𝑒,𝑓,𝑔,𝑖,𝑗,𝑏,𝑑) 𝑊(𝑔,𝑏) 𝑌(𝑒,𝑓,𝑔,𝑖,𝑗,𝑏,𝑑) 𝑍(𝑒,𝑓,𝑖,𝑗,𝑏,𝑑)

**Proof of Theorem cdlemkyu**
Step	Hyp	Ref	Expression
1		cdlemk5.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemk5.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdlemk5.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdlemk5.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdlemk5.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemk5.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemk5.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemk5.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
9		cdlemk5.z	. . 3 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ^◡𝐹))))
10		cdlemk5.y	. . 3 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
11		cdlemk5b.s	. . 3 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
12		cdlemk5b.u1	. . 3 ⊢ 𝑉 = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑑))))))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemky 41296	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)))) → ⦋𝐺 / 𝑔⦌𝑌 = ((𝑏𝑉𝐺)‘𝑃))
14		simp3l 1203	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)))) → 𝑏 ∈ 𝑇)
15		simp13l 1290	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)))) → 𝐺 ∈ 𝑇)
16		cdlemk5.o2	. . . . 5 ⊢ 𝑄 = (𝑆‘𝑏)
17		cdlemk5.u2	. . . . 5 ⊢ 𝐶 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑏))))))
18	1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 16, 17	cdlemkuu 41265	. . . 4 ⊢ ((𝑏 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑏𝑉𝐺) = (𝐶‘𝐺))
19	14, 15, 18	syl2anc 585	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)))) → (𝑏𝑉𝐺) = (𝐶‘𝐺))
20	19	fveq1d 6844	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)))) → ((𝑏𝑉𝐺)‘𝑃) = ((𝐶‘𝐺)‘𝑃))
21	13, 20	eqtrd 2772	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐺)))) → ⦋𝐺 / 𝑔⦌𝑌 = ((𝐶‘𝐺)‘𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ⦋csb 3851 class class class wbr 5100 ↦ cmpt 5181 I cid 5526 ^◡ccnv 5631 ↾ cres 5634 ∘ ccom 5636 ‘cfv 6500 ℩crio 7324 (class class class)co 7368 ∈ cmpo 7370 Basecbs 17148 lecple 17196 joincjn 18246 meetcmee 18247 Atomscatm 39633 HLchlt 39720 LHypclh 40354 LTrncltrn 40471 trLctrl 40528

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-riotaBAD 39323

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-undef 8225 df-map 8777 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-oposet 39546 df-ol 39548 df-oml 39549 df-covers 39636 df-ats 39637 df-atl 39668 df-cvlat 39692 df-hlat 39721 df-llines 39868 df-lplanes 39869 df-lvols 39870 df-lines 39871 df-psubsp 39873 df-pmap 39874 df-padd 40166 df-lhyp 40358 df-laut 40359 df-ldil 40474 df-ltrn 40475 df-trl 40529

This theorem is referenced by: cdlemkyuu 41298

W3C validator