Theorem bj-inftyexpiinj 37701

Description: Injectivity of the parameterization +∞eⁱ. Remark: a more conceptual proof would use bj-inftyexpiinv 37700 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 ↔ (+∞ei‘𝐴) = (+∞ei‘𝐵)))

Proof of Theorem bj-inftyexpiinj

Step	Hyp	Ref	Expression
1		fveq2 6867	. 2 ⊢ (𝐴 = 𝐵 → (+∞ei‘𝐴) = (+∞ei‘𝐵))
2		fveq2 6867	. . 3 ⊢ ((+∞ei‘𝐴) = (+∞ei‘𝐵) → (1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)))
3		bj-inftyexpiinv 37700	. . . . . . 7 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴)
4	3	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei‘𝐴)) = 𝐴)
5	4	eqeq1d 2764	. . . . 5 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)) ↔ 𝐴 = (1st ‘(+∞ei‘𝐵))))
6	5	biimpd 231	. . . 4 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)) → 𝐴 = (1st ‘(+∞ei‘𝐵))))
7		bj-inftyexpiinv 37700	. . . . . 6 ⊢ (𝐵 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐵)) = 𝐵)
8	7	adantl 485	. . . . 5 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei‘𝐵)) = 𝐵)
9	8	eqeq2d 2773	. . . 4 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1st ‘(+∞ei‘𝐵)) ↔ 𝐴 = 𝐵))
10	6, 9	sylibd 241	. . 3 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)) → 𝐴 = 𝐵))
11	2, 10	syl5 34	. 2 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((+∞ei‘𝐴) = (+∞ei‘𝐵) → 𝐴 = 𝐵))
12	1, 11	impbid2 228	1 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (+∞ei‘𝐴) = (+∞ei‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ‘cfv 6521  (class class class)co 7396  1^st c1st 7968  -cneg 11415  (,]cioc 13350  πcpi 16096  +∞eⁱcinftyexpi 37698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718  ax-cnex 11129

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fv 6529  df-1st 7970  df-bj-inftyexpi 37699

This theorem is referenced by:  bj-pinftynminfty  37719