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Theorem bj-inftyexpiinj 34371
Description: Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 34370 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 ↔ (+∞ei𝐴) = (+∞ei𝐵)))

Proof of Theorem bj-inftyexpiinj
StepHypRef Expression
1 fveq2 6666 . 2 (𝐴 = 𝐵 → (+∞ei𝐴) = (+∞ei𝐵))
2 fveq2 6666 . . 3 ((+∞ei𝐴) = (+∞ei𝐵) → (1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)))
3 bj-inftyexpiinv 34370 . . . . . . 7 (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei𝐴)) = 𝐴)
43adantr 481 . . . . . 6 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei𝐴)) = 𝐴)
54eqeq1d 2826 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)) ↔ 𝐴 = (1st ‘(+∞ei𝐵))))
65biimpd 230 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)) → 𝐴 = (1st ‘(+∞ei𝐵))))
7 bj-inftyexpiinv 34370 . . . . . 6 (𝐵 ∈ (-π(,]π) → (1st ‘(+∞ei𝐵)) = 𝐵)
87adantl 482 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei𝐵)) = 𝐵)
98eqeq2d 2835 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1st ‘(+∞ei𝐵)) ↔ 𝐴 = 𝐵))
106, 9sylibd 240 . . 3 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)) → 𝐴 = 𝐵))
112, 10syl5 34 . 2 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((+∞ei𝐴) = (+∞ei𝐵) → 𝐴 = 𝐵))
121, 11impbid2 227 1 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (+∞ei𝐴) = (+∞ei𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2106  cfv 6351  (class class class)co 7151  1st c1st 7681  -cneg 10863  (,]cioc 12732  πcpi 15412  +∞eicinftyexpi 34368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fv 6359  df-1st 7683  df-bj-inftyexpi 34369
This theorem is referenced by:  bj-pinftynminfty  34389
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