Theorem bj-inftyexpiinj 38398

Description: Injectivity of the parameterization +∞eⁱ. Remark: a more conceptual proof would use bj-inftyexpiinv 38397 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 6916	. 2 ⊢ (𝐴 = 𝐵 → (+∞ei‘𝐴) = (+∞ei‘𝐵))
2		fveq2 6916	. . 3 ⊢ ((+∞ei‘𝐴) = (+∞ei‘𝐵) → (1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)))
3		bj-inftyexpiinv 38397	. . . . . . 7 ⊢ (𝐴 ∈ ( -π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴)
4	3	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ ( -π(,]π) ∧ 𝐵 ∈ ( -π(,]π)) → (1st ‘(+∞ei‘𝐴)) = 𝐴)
5	4	eqeq1d 2765	. . . . 5 ⊢ ((𝐴 ∈ ( -π(,]π) ∧ 𝐵 ∈ ( -π(,]π)) → ((1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)) ↔ 𝐴 = (1st ‘(+∞ei‘𝐵))))
6	5	biimpd 232	. . . 4 ⊢ ((𝐴 ∈ ( -π(,]π) ∧ 𝐵 ∈ ( -π(,]π)) → ((1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)) → 𝐴 = (1st ‘(+∞ei‘𝐵))))
7		bj-inftyexpiinv 38397	. . . . . 6 ⊢ (𝐵 ∈ ( -π(,]π) → (1st ‘(+∞ei‘𝐵)) = 𝐵)
8	7	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ ( -π(,]π) ∧ 𝐵 ∈ ( -π(,]π)) → (1st ‘(+∞ei‘𝐵)) = 𝐵)
9	8	eqeq2d 2774	. . . 4 ⊢ ((𝐴 ∈ ( -π(,]π) ∧ 𝐵 ∈ ( -π(,]π)) → (𝐴 = (1st ‘(+∞ei‘𝐵)) ↔ 𝐴 = 𝐵))
10	6, 9	sylibd 242	. . 3 ⊢ ((𝐴 ∈ ( -π(,]π) ∧ 𝐵 ∈ ( -π(,]π)) → ((1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)) → 𝐴 = 𝐵))
11	2, 10	syl5 35	. 2 ⊢ ((𝐴 ∈ ( -π(,]π) ∧ 𝐵 ∈ ( -π(,]π)) → ((+∞ei‘𝐴) = (+∞ei‘𝐵) → 𝐴 = 𝐵))
12	1, 11	impbid2 229	1 ⊢ ((𝐴 ∈ ( -π(,]π) ∧ 𝐵 ∈ ( -π(,]π)) → (𝐴 = 𝐵 ↔ (+∞ei‘𝐴) = (+∞ei‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033   -cneg 11581  (,]cioc 13517  πcpi 16276  +∞eⁱcinftyexpi 38395

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783  ax-cnex 11295

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-1st 8035  df-bj-inftyexpi 38396

This theorem is used by:  bj-pinftynminfty  38416