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Theorem bj-inftyexpiinj 36090
Description: Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 36089 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 6892 . 2 (𝐴 = 𝐵 → (+∞ei𝐴) = (+∞ei𝐵))
2 fveq2 6892 . . 3 ((+∞ei𝐴) = (+∞ei𝐵) → (1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)))
3 bj-inftyexpiinv 36089 . . . . . . 7 (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei𝐴)) = 𝐴)
43adantr 482 . . . . . 6 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei𝐴)) = 𝐴)
54eqeq1d 2735 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)) ↔ 𝐴 = (1st ‘(+∞ei𝐵))))
65biimpd 228 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei𝐴)) = (1st ‘(+∞ei𝐵)) → 𝐴 = (1st ‘(+∞ei𝐵))))
7 bj-inftyexpiinv 36089 . . . . . 6 (𝐵 ∈ (-π(,]π) → (1st ‘(+∞ei𝐵)) = 𝐵)
87adantl 483 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei𝐵)) = 𝐵)
98eqeq2d 2744 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1st ‘(+∞ei𝐵)) ↔ 𝐴 = 𝐵))
106, 9sylibd 238 . . 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 225 1 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (+∞ei𝐴) = (+∞ei𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cfv 6544  (class class class)co 7409  1st c1st 7973  -cneg 11445  (,]cioc 13325  πcpi 16010  +∞eicinftyexpi 36087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-cnex 11166
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-bj-inftyexpi 36088
This theorem is referenced by:  bj-pinftynminfty  36108
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