Theorem bj-inftyexpiinj 38130

Description: Injectivity of the parameterization +∞eⁱ. Remark: a more conceptual proof would use bj-inftyexpiinv 38129 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 6885	. 2 ⊢ (𝐴 = 𝐵 → (+∞ei‘𝐴) = (+∞ei‘𝐵))
2		fveq2 6885	. . 3 ⊢ ((+∞ei‘𝐴) = (+∞ei‘𝐵) → (1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)))
3		bj-inftyexpiinv 38129	. . . . . . 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 2763	. . . . 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 38129	. . . . . 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 2772	. . . 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 2145  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  -cneg 11542  (,]cioc 13477  πcpi 16232  +∞eⁱcinftyexpi 38127

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751  ax-cnex 11256

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 8001  df-bj-inftyexpi 38128

This theorem is used by:  bj-pinftynminfty  38148