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Theorem bj-inftyexpiinj 32273
 Description: Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 32272 and the fact that a function with a retraction is injective. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-inftyexpiinj ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))

Proof of Theorem bj-inftyexpiinj
StepHypRef Expression
1 fveq2 6103 . 2 (𝐴 = 𝐵 → (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵))
2 fveq2 6103 . . 3 ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → (1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)))
3 bj-inftyexpiinv 32272 . . . . . . 7 (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)
43adantr 480 . . . . . 6 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)
54eqeq1d 2612 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = (1st ‘(inftyexpi ‘𝐵))))
65biimpd 218 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)) → 𝐴 = (1st ‘(inftyexpi ‘𝐵))))
7 bj-inftyexpiinv 32272 . . . . . 6 (𝐵 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐵)) = 𝐵)
87adantl 481 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(inftyexpi ‘𝐵)) = 𝐵)
98eqeq2d 2620 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = 𝐵))
106, 9sylibd 228 . . 3 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)) → 𝐴 = 𝐵))
112, 10syl5 33 . 2 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → 𝐴 = 𝐵))
121, 11impbid2 215 1 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  -cneg 10146  (,]cioc 12047  πcpi 14636  inftyexpi cinftyexpi 32270 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-bj-inftyexpi 32271 This theorem is referenced by:  bj-pinftynminfty  32291
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