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Theorem bj-inftyexpiinj 34990
Description: Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 34989 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 6668 . 2 (𝐴 = 𝐵 → (+∞ei𝐴) = (+∞ei𝐵))
2 fveq2 6668 . . 3 ((+∞ei𝐴) = (+∞ei𝐵) → (1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)))
3 bj-inftyexpiinv 34989 . . . . . . 7 (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei𝐴)) = 𝐴)
43adantr 484 . . . . . 6 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei𝐴)) = 𝐴)
54eqeq1d 2740 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)) ↔ 𝐴 = (1st ‘(+∞ei𝐵))))
65biimpd 232 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)) → 𝐴 = (1st ‘(+∞ei𝐵))))
7 bj-inftyexpiinv 34989 . . . . . 6 (𝐵 ∈ (-π(,]π) → (1st ‘(+∞ei𝐵)) = 𝐵)
87adantl 485 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei𝐵)) = 𝐵)
98eqeq2d 2749 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1st ‘(+∞ei𝐵)) ↔ 𝐴 = 𝐵))
106, 9sylibd 242 . . 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 229 1 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (+∞ei𝐴) = (+∞ei𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  cfv 6333  (class class class)co 7164  1st c1st 7705  -cneg 10942  (,]cioc 12815  πcpi 15505  +∞eicinftyexpi 34987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473  ax-cnex 10664
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6291  df-fun 6335  df-fv 6341  df-1st 7707  df-bj-inftyexpi 34988
This theorem is referenced by:  bj-pinftynminfty  35008
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