Theorem bj-inftyexpiinj 38267

Description: Injectivity of the parameterization +∞eⁱ. Remark: a more conceptual proof would use bj-inftyexpiinv 38266 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 6890	. 2 ⊢ (𝐴 = 𝐵 → (+∞ei‘𝐴) = (+∞ei‘𝐵))
2		fveq2 6890	. . 3 ⊢ ((+∞ei‘𝐴) = (+∞ei‘𝐵) → (1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵)))
3		bj-inftyexpiinv 38266	. . . . . . 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 2762	. . . . 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 38266	. . . . . 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 2771	. . . 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 1569   ∈ wcel 2144  ‘cfv 6543  (class class class)co 7424  1^st c1st 8003   -cneg 11549  (,]cioc 13484  πcpi 16241  +∞eⁱcinftyexpi 38264

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755  ax-cnex 11263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6497  df-fun 6545  df-fv 6551  df-1st 8005  df-bj-inftyexpi 38265

This theorem is used by:  bj-pinftynminfty  38285