Theorem cdlemksv 40827

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 6907	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑅‘𝑓) = (𝑅‘𝐺))
2	1	oveq2d 7447	. . . . 5 ⊢ (𝑓 = 𝐺 → (𝑃 ∨ (𝑅‘𝑓)) = (𝑃 ∨ (𝑅‘𝐺)))
3		coeq1 5871	. . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓 ∘ ◡𝐹) = (𝐺 ∘ ◡𝐹))
4	3	fveq2d 6911	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑅‘(𝑓 ∘ ◡𝐹)) = (𝑅‘(𝐺 ∘ ◡𝐹)))
5	4	oveq2d 7447	. . . . 5 ⊢ (𝑓 = 𝐺 → ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))) = ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))
6	2, 5	oveq12d 7449	. . . 4 ⊢ (𝑓 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹)))))
7	6	eqeq2d 2746	. . 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 7390	. 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 7392	. 2 ⊢ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))) ∈ V
11	8, 9, 10	fvmpt 7016	1 ⊢ (𝐺 ∈ 𝑇 → (𝑆‘𝐺) = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ↦ cmpt 5231  ^◡ccnv 5688   ∘ ccom 5693  ‘cfv 6563  ℩crio 7387  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Atomscatm 39245  LHypclh 39967  LTrncltrn 40084  trLctrl 40141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434

This theorem is referenced by:  cdlemksel  40828  cdlemksv2  40830  cdlemkuvN  40847  cdlemkuel  40848  cdlemkuv2  40850  cdlemkuv-2N  40866  cdlemkuu  40878