Theorem cdlemksv 41678

Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.)

Hypotheses

Ref	Expression
cdlemk.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk.l	⊢ ≤ = (le‘𝐾)
cdlemk.j	⊢ ∨ = (join‘𝐾)
cdlemk.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk.m	⊢ ∧ = (meet‘𝐾)
cdlemk.s	⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))))))

Assertion

Ref	Expression
cdlemksv	⊢ (𝐺 ∈ 𝑇 → (𝑆‘𝐺) = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))))

Distinct variable groups:   ∧ ,𝑓   ∨ ,𝑓   𝑓,𝐹   𝑓,𝑖,𝐺   𝑓,𝑁   𝑃,𝑓   𝑅,𝑓   𝑇,𝑓   𝑓,𝑊

Allowed substitution hints:   𝐴(𝑓, 𝑖)   𝐵(𝑓, 𝑖)   𝑃(𝑖)   𝑅(𝑖)   𝑆(𝑓, 𝑖)   𝑇(𝑖)   𝐹(𝑖)   𝐻(𝑓, 𝑖)   ∨ (𝑖)   𝐾(𝑓, 𝑖)   ≤ (𝑓, 𝑖)   ∧ (𝑖)   𝑁(𝑖)   𝑊(𝑖)

Proof of Theorem cdlemksv

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑅‘𝑓) = (𝑅‘𝐺))
2	1	oveq2d 7435	. . . . 5 ⊢ (𝑓 = 𝐺 → (𝑃 ∨ (𝑅‘𝑓)) = (𝑃 ∨ (𝑅‘𝐺)))
3		coeq1 5845	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓 ∘ ◡𝐹) = (𝐺 ∘ ◡𝐹))
4	3	fveq2d 6889	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑅‘(𝑓 ∘ ◡𝐹)) = (𝑅‘(𝐺 ∘ ◡𝐹)))
5	4	oveq2d 7435	. . . . 5 ⊢ (𝑓 = 𝐺 → ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))) = ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))
6	2, 5	oveq12d 7437	. . . 4 ⊢ (𝑓 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹)))))
7	6	eqeq2d 2776	. . 3 ⊢ (𝑓 = 𝐺 → ((𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))) ↔ (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))))
8	7	riotabidv 7378	. 2 ⊢ (𝑓 = 𝐺 → (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))))) = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))))
9		cdlemk.s	. 2 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))))))
10		riotaex 7380	. 2 ⊢ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))) ∈ V
11	8, 9, 10	fvmpt 6993	1 ⊢ (𝐺 ∈ 𝑇 → (𝑆‘𝐺) = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ↦ cmpt 5194  ^◡ccnv 5662   ∘ ccom 5667  ‘cfv 6540  ℩crio 7375  (class class class)co 7419  Basecbs 17293  lecple 17341  joincjn 18391  meetcmee 18392  Atomscatm 40097  LHypclh 40818  LTrncltrn 40935  trLctrl 40992

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-riota 7376  df-ov 7422

This theorem is used by:  cdlemksel  41679  cdlemksv2  41681  cdlemkuvN  41698  cdlemkuel  41699  cdlemkuv2  41701  cdlemkuv-2N  41717  cdlemkuu  41729