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Theorem bj-inftyexpiinj 33226
 Description: Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 33225 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 6229 . 2 (𝐴 = 𝐵 → (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵))
2 fveq2 6229 . . 3 ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → (1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)))
3 bj-inftyexpiinv 33225 . . . . . . 7 (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)
43adantr 480 . . . . . 6 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)
54eqeq1d 2653 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = (1st ‘(inftyexpi ‘𝐵))))
65biimpd 219 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)) → 𝐴 = (1st ‘(inftyexpi ‘𝐵))))
7 bj-inftyexpiinv 33225 . . . . . 6 (𝐵 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐵)) = 𝐵)
87adantl 481 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(inftyexpi ‘𝐵)) = 𝐵)
98eqeq2d 2661 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = 𝐵))
106, 9sylibd 229 . . 3 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)) → 𝐴 = 𝐵))
112, 10syl5 34 . 2 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → 𝐴 = 𝐵))
121, 11impbid2 216 1 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  -cneg 10305  (,]cioc 12214  πcpi 14841  inftyexpi cinftyexpi 33223 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-1st 7210  df-bj-inftyexpi 33224 This theorem is referenced by:  bj-pinftynminfty  33244
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