Theorem cdlemksv 36132

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 6191	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑅‘𝑓) = (𝑅‘𝐺))
2	1	oveq2d 6666	. . . . 5 ⊢ (𝑓 = 𝐺 → (𝑃 ∨ (𝑅‘𝑓)) = (𝑃 ∨ (𝑅‘𝐺)))
3		coeq1 5279	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓 ∘ ◡𝐹) = (𝐺 ∘ ◡𝐹))
4	3	fveq2d 6195	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑅‘(𝑓 ∘ ◡𝐹)) = (𝑅‘(𝐺 ∘ ◡𝐹)))
5	4	oveq2d 6666	. . . . 5 ⊢ (𝑓 = 𝐺 → ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))) = ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))
6	2, 5	oveq12d 6668	. . . 4 ⊢ (𝑓 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹)))))
7	6	eqeq2d 2632	. . 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 6613	. 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 6615	. 2 ⊢ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))) ∈ V
11	8, 9, 10	fvmpt 6282	1 ⊢ (𝐺 ∈ 𝑇 → (𝑆‘𝐺) = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ↦ cmpt 4729  ^◡ccnv 5113   ∘ ccom 5118  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  Basecbs 15857  lecple 15948  joincjn 16944  meetcmee 16945  Atomscatm 34550  LHypclh 35270  LTrncltrn 35387  trLctrl 35445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653

This theorem is referenced by:  cdlemksel  36133  cdlemksv2  36135  cdlemkuvN  36152  cdlemkuel  36153  cdlemkuv2  36155  cdlemkuv-2N  36171  cdlemkuu  36183