bj-inftyexpiinj - Mathbox for BJ

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Theorem bj-inftyexpiinj 33226

Description: Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 33225 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 6229	. 2 ⊢ (𝐴 = 𝐵 → (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵))
2		fveq2 6229	. . 3 ⊢ ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → (1^st ‘(inftyexpi ‘𝐴)) = (1^st ‘(inftyexpi ‘𝐵)))
3		bj-inftyexpiinv 33225	. . . . . . 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 2653	. . . . 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 33225	. . . . . 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 2661	. . . 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 383 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 1^st c1st 7208 -cneg 10305 (,]cioc 12214 πcpi 14841 inftyexpi cinftyexpi 33223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-iota 5889 df-fun 5928 df-fv 5934 df-1st 7210 df-bj-inftyexpi 33224

This theorem is referenced by: bj-pinftynminfty 33244

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