bj-inftyexpiinj - Mathbox for BJ

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Theorem bj-inftyexpiinj 32273

Description: Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 32272 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 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))

**Proof of Theorem bj-inftyexpiinj**
Step	Hyp	Ref	Expression
1		fveq2 6103	. 2 ⊢ (𝐴 = 𝐵 → (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵))
2		fveq2 6103	. . 3 ⊢ ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → (1^st ‘(inftyexpi ‘𝐴)) = (1^st ‘(inftyexpi ‘𝐵)))
3		bj-inftyexpiinv 32272	. . . . . . 7 ⊢ (𝐴 ∈ (-π(,]π) → (1^st ‘(inftyexpi ‘𝐴)) = 𝐴)
4	3	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1^st ‘(inftyexpi ‘𝐴)) = 𝐴)
5	4	eqeq1d 2612	. . . . 5 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1^st ‘(inftyexpi ‘𝐴)) = (1^st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = (1^st ‘(inftyexpi ‘𝐵))))
6	5	biimpd 218	. . . 4 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1^st ‘(inftyexpi ‘𝐴)) = (1^st ‘(inftyexpi ‘𝐵)) → 𝐴 = (1^st ‘(inftyexpi ‘𝐵))))
7		bj-inftyexpiinv 32272	. . . . . 6 ⊢ (𝐵 ∈ (-π(,]π) → (1^st ‘(inftyexpi ‘𝐵)) = 𝐵)
8	7	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1^st ‘(inftyexpi ‘𝐵)) = 𝐵)
9	8	eqeq2d 2620	. . . 4 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1^st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = 𝐵))
10	6, 9	sylibd 228	. . 3 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1^st ‘(inftyexpi ‘𝐴)) = (1^st ‘(inftyexpi ‘𝐵)) → 𝐴 = 𝐵))
11	2, 10	syl5 33	. 2 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → 𝐴 = 𝐵))
12	1, 11	impbid2 215	1 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 -cneg 10146 (,]cioc 12047 πcpi 14636 inftyexpi cinftyexpi 32270

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-bj-inftyexpi 32271

This theorem is referenced by: bj-pinftynminfty 32291

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