Theorem bj-inftyexpiinj 35380

Description: Injectivity of the parameterization +∞eⁱ. Remark: a more conceptual proof would use bj-inftyexpiinv 35379 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 6774	. 2 ⊢ (𝐴 = 𝐵 → (+∞ei‘𝐴) = (+∞ei‘𝐵))
2		fveq2 6774	. . 3 ⊢ ((+∞ei‘𝐴) = (+∞ei‘𝐵) → (1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)))
3		bj-inftyexpiinv 35379	. . . . . . 7 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴)
4	3	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei‘𝐴)) = 𝐴)
5	4	eqeq1d 2740	. . . . 5 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)) ↔ 𝐴 = (1st ‘(+∞ei‘𝐵))))
6	5	biimpd 228	. . . 4 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)) → 𝐴 = (1st ‘(+∞ei‘𝐵))))
7		bj-inftyexpiinv 35379	. . . . . 6 ⊢ (𝐵 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐵)) = 𝐵)
8	7	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei‘𝐵)) = 𝐵)
9	8	eqeq2d 2749	. . . 4 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1st ‘(+∞ei‘𝐵)) ↔ 𝐴 = 𝐵))
10	6, 9	sylibd 238	. . 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 225	1 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (+∞ei‘𝐴) = (+∞ei‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  -cneg 11206  (,]cioc 13080  πcpi 15776  +∞eⁱcinftyexpi 35377

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-cnex 10927

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831  df-bj-inftyexpi 35378

This theorem is referenced by:  bj-pinftynminfty  35398