cdlemkyu - Mathbox for Norm Megill

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Theorem cdlemkyu 41183

Description: Convert between function and explicit forms. 𝐶 represents 𝑍 in cdlemkuu 41151. 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 41182	. 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 1202	. . . 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 1289	. . . 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 41151	. . . 4 ⊢ ((𝑏 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑏𝑉𝐺) = (𝐶‘𝐺))
19	14, 15, 18	syl2anc 584	. . 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 6836	. 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 2771	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 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ⦋csb 3849 class class class wbr 5098 ↦ cmpt 5179 I cid 5518 ^◡ccnv 5623 ↾ cres 5626 ∘ ccom 5628 ‘cfv 6492 ℩crio 7314 (class class class)co 7358 ∈ cmpo 7360 Basecbs 17136 lecple 17184 joincjn 18234 meetcmee 18235 Atomscatm 39519 HLchlt 39606 LHypclh 40240 LTrncltrn 40357 trLctrl 40414

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-riotaBAD 39209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-undef 8215 df-map 8765 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-oposet 39432 df-ol 39434 df-oml 39435 df-covers 39522 df-ats 39523 df-atl 39554 df-cvlat 39578 df-hlat 39607 df-llines 39754 df-lplanes 39755 df-lvols 39756 df-lines 39757 df-psubsp 39759 df-pmap 39760 df-padd 40052 df-lhyp 40244 df-laut 40245 df-ldil 40360 df-ltrn 40361 df-trl 40415

This theorem is referenced by: cdlemkyuu 41184

W3C validator