bj-inftyexpiinj - Mathbox for BJ

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Theorem bj-inftyexpiinj 33096

Description: Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 33095 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 6191	. 2 ⊢ (𝐴 = 𝐵 → (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵))
2		fveq2 6191	. . 3 ⊢ ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → (1^st ‘(inftyexpi ‘𝐴)) = (1^st ‘(inftyexpi ‘𝐵)))
3		bj-inftyexpiinv 33095	. . . . . . 7 ⊢ (𝐴 ∈ (-π(,]π) → (1^st ‘(inftyexpi ‘𝐴)) = 𝐴)
4	3	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1^st ‘(inftyexpi ‘𝐴)) = 𝐴)
5	4	eqeq1d 2624	. . . . 5 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1^st ‘(inftyexpi ‘𝐴)) = (1^st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = (1^st ‘(inftyexpi ‘𝐵))))
6	5	biimpd 219	. . . 4 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1^st ‘(inftyexpi ‘𝐴)) = (1^st ‘(inftyexpi ‘𝐵)) → 𝐴 = (1^st ‘(inftyexpi ‘𝐵))))
7		bj-inftyexpiinv 33095	. . . . . 6 ⊢ (𝐵 ∈ (-π(,]π) → (1^st ‘(inftyexpi ‘𝐵)) = 𝐵)
8	7	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1^st ‘(inftyexpi ‘𝐵)) = 𝐵)
9	8	eqeq2d 2632	. . . 4 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1^st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = 𝐵))
10	6, 9	sylibd 229	. . 3 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1^st ‘(inftyexpi ‘𝐴)) = (1^st ‘(inftyexpi ‘𝐵)) → 𝐴 = 𝐵))
11	2, 10	syl5 34	. 2 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → 𝐴 = 𝐵))
12	1, 11	impbid2 216	1 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 -cneg 10267 (,]cioc 12176 πcpi 14797 inftyexpi cinftyexpi 33093

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-bj-inftyexpi 33094

This theorem is referenced by: bj-pinftynminfty 33114

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