Theorem lautle 41407

Description: Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Hypotheses

Ref	Expression
lautset.b	⊢ 𝐵 = (Base‘𝐾)
lautset.l	⊢ ≤ = (le‘𝐾)
lautset.i	⊢ 𝐼 = (LAut‘𝐾)

Assertion

Ref	Expression
lautle	⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))

Proof of Theorem lautle

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lautset.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		lautset.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		lautset.i	. . . 4 ⊢ 𝐼 = (LAut‘𝐾)
4	1, 2, 3	islaut 41406	. . 3 ⊢ (𝐾 ∈ 𝑉 → (𝐹 ∈ 𝐼 ↔ (𝐹:𝐵⟶1-1onto𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)))))
5	4	simplbda 505	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
6		breq1 5111	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
7		fveq2 6916	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
8	7	breq1d 5118	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑦)))
9	6, 8	bibi12d 348	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑦 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑦))))
10		breq2 5112	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
11		fveq2 6916	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
12	11	breq2d 5120	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
13	10, 12	bibi12d 348	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))))
14	9, 13	rspc2v 3590	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))))
15	5, 14	mpan9 516	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   class class class wbr 5108  ⟶1-1ontowf1o 6565  ‘cfv 6566  Basecbs 17431  lecple 17479  LAutclaut 41308

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-laut 41312

This theorem is used by:  lautcnvle  41412  lautlt  41414  lautj  41416  lautm  41417  lauteq  41418  lautco  41420  ltrnle  41452