Theorem cdlemkuu 40896

Description: Convert between function and operation forms of 𝑌. TODO: Use operation form everywhere. (Contributed by NM, 6-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk3.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk3.l	⊢ ≤ = (le‘𝐾)
cdlemk3.j	⊢ ∨ = (join‘𝐾)
cdlemk3.m	⊢ ∧ = (meet‘𝐾)
cdlemk3.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk3.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk3.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk3.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk3.s	⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))))))
cdlemk3.u1	⊢ 𝑌 = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝑑))))))
cdlemk3.o2	⊢ 𝑄 = (𝑆‘𝐷)
cdlemk3.u2	⊢ 𝑍 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))

Assertion

Ref	Expression
cdlemkuu	⊢ ((𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐷𝑌𝐺) = (𝑍‘𝐺))

Distinct variable groups:   𝑒,𝑑,𝑓,𝑖, ∧   ≤ ,𝑖   ∨ ,𝑑,𝑒,𝑓,𝑖   𝐴,𝑖   𝑗,𝑑,𝐷,𝑒,𝑓,𝑖   𝑓,𝐹,𝑖   𝐺,𝑑,𝑒,𝑗   𝑖,𝐻   𝑖,𝐾   𝑓,𝑁,𝑖   𝑃,𝑑,𝑒,𝑓,𝑖   𝑄,𝑑,𝑒   𝑅,𝑑,𝑒,𝑓,𝑖   𝑇,𝑑,𝑒,𝑓,𝑖   𝑊,𝑑,𝑒,𝑓,𝑖

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

Proof of Theorem cdlemkuu

Step	Hyp	Ref	Expression
1		fveq2 6861	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑆‘𝑑) = (𝑆‘𝐷))
2		cdlemk3.o2	. . . . . . . . 9 ⊢ 𝑄 = (𝑆‘𝐷)
3	1, 2	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑆‘𝑑) = 𝑄)
4	3	fveq1d 6863	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑆‘𝑑)‘𝑃) = (𝑄‘𝑃))
5		cnveq 5840	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ◡𝑑 = ◡𝐷)
6	5	coeq2d 5829	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑒 ∘ ◡𝑑) = (𝑒 ∘ ◡𝐷))
7	6	fveq2d 6865	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑅‘(𝑒 ∘ ◡𝑑)) = (𝑅‘(𝑒 ∘ ◡𝐷)))
8	4, 7	oveq12d 7408	. . . . . 6 ⊢ (𝑑 = 𝐷 → (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝑑))) = ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))
9	8	oveq2d 7406	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝑑)))) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷)))))
10	9	eqeq2d 2741	. . . 4 ⊢ (𝑑 = 𝐷 → ((𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝑑)))) ↔ (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))
11	10	riotabidv 7349	. . 3 ⊢ (𝑑 = 𝐷 → (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝑑))))) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))
12		fveq2 6861	. . . . . . 7 ⊢ (𝑒 = 𝐺 → (𝑅‘𝑒) = (𝑅‘𝐺))
13	12	oveq2d 7406	. . . . . 6 ⊢ (𝑒 = 𝐺 → (𝑃 ∨ (𝑅‘𝑒)) = (𝑃 ∨ (𝑅‘𝐺)))
14		coeq1 5824	. . . . . . . 8 ⊢ (𝑒 = 𝐺 → (𝑒 ∘ ◡𝐷) = (𝐺 ∘ ◡𝐷))
15	14	fveq2d 6865	. . . . . . 7 ⊢ (𝑒 = 𝐺 → (𝑅‘(𝑒 ∘ ◡𝐷)) = (𝑅‘(𝐺 ∘ ◡𝐷)))
16	15	oveq2d 7406	. . . . . 6 ⊢ (𝑒 = 𝐺 → ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))) = ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))
17	13, 16	oveq12d 7408	. . . . 5 ⊢ (𝑒 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷)))))
18	17	eqeq2d 2741	. . . 4 ⊢ (𝑒 = 𝐺 → ((𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷)))) ↔ (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))))
19	18	riotabidv 7349	. . 3 ⊢ (𝑒 = 𝐺 → (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))))
20		cdlemk3.u1	. . 3 ⊢ 𝑌 = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝑑))))))
21		riotaex 7351	. . 3 ⊢ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))) ∈ V
22	11, 19, 20, 21	ovmpo 7552	. 2 ⊢ ((𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐷𝑌𝐺) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))))
23		cdlemk3.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
24		cdlemk3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
25		cdlemk3.j	. . . 4 ⊢ ∨ = (join‘𝐾)
26		cdlemk3.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
27		cdlemk3.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
28		cdlemk3.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
29		cdlemk3.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
30		cdlemk3.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
31		cdlemk3.u2	. . . 4 ⊢ 𝑍 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))
32	23, 24, 25, 26, 27, 28, 29, 30, 31	cdlemksv 40845	. . 3 ⊢ (𝐺 ∈ 𝑇 → (𝑍‘𝐺) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))))
33	32	adantl 481	. 2 ⊢ ((𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑍‘𝐺) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))))
34	22, 33	eqtr4d 2768	1 ⊢ ((𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐷𝑌𝐺) = (𝑍‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5191  ^◡ccnv 5640   ∘ ccom 5645  ‘cfv 6514  ℩crio 7346  (class class class)co 7390   ∈ cmpo 7392  Basecbs 17186  lecple 17234  joincjn 18279  meetcmee 18280  Atomscatm 39263  LHypclh 39985  LTrncltrn 40102  trLctrl 40159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395

This theorem is referenced by:  cdlemk31  40897  cdlemkuel-3  40899  cdlemkuv2-3N  40900  cdlemk18-3N  40901  cdlemk22-3  40902  cdlemkyu  40928