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Theorem ntrk1k3eqk13 38819
 Description: An interior function is both monotone and sub-linear if and only if it is finitely linear. (Contributed by RP, 18-Jun-2021.)
Assertion
Ref Expression
ntrk1k3eqk13 ((∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)))
Distinct variable groups:   𝐵,𝑠,𝑡   𝐼,𝑠,𝑡

Proof of Theorem ntrk1k3eqk13
StepHypRef Expression
1 r19.26-2 3191 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ∧ ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))) ↔ (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))))
2 eqss 3747 . . 3 ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ∧ ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))))
322ralbii 3107 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ∧ ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))))
4 isotone2 38818 . . 3 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)))
54anbi1i 733 . 2 ((∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))) ↔ (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))))
61, 3, 53bitr4ri 293 1 ((∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1620  ∀wral 3038   ∩ cin 3702   ⊆ wss 3703  𝒫 cpw 4290  ‘cfv 6037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-iota 6000  df-fv 6045 This theorem is referenced by: (None)
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