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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-inftyexpiinj | Structured version Visualization version GIF version |
Description: Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 34623 and the fact that a function with a retraction is injective. (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
bj-inftyexpiinj | ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (+∞ei‘𝐴) = (+∞ei‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6645 | . 2 ⊢ (𝐴 = 𝐵 → (+∞ei‘𝐴) = (+∞ei‘𝐵)) | |
2 | fveq2 6645 | . . 3 ⊢ ((+∞ei‘𝐴) = (+∞ei‘𝐵) → (1st ‘(+∞ei‘𝐴)) = (1st ‘(+∞ei‘𝐵))) | |
3 | bj-inftyexpiinv 34623 | . . . . . . 7 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐴)) = 𝐴) | |
4 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei‘𝐴)) = 𝐴) |
5 | 4 | eqeq1d 2800 | . . . . 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 34623 | . . . . . 6 ⊢ (𝐵 ∈ (-π(,]π) → (1st ‘(+∞ei‘𝐵)) = 𝐵) | |
8 | 7 | adantl 485 | . . . . 5 ⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(+∞ei‘𝐵)) = 𝐵) |
9 | 8 | eqeq2d 2809 | . . . 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 34 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 -cneg 10860 (,]cioc 12727 πcpi 15412 +∞eicinftyexpi 34621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-1st 7671 df-bj-inftyexpi 34622 |
This theorem is referenced by: bj-pinftynminfty 34642 |
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