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Mirrors > Home > ILE Home > Th. List > reef11 | GIF version |
Description: The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.) |
Ref | Expression |
---|---|
reef11 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((exp‘𝐴) = (exp‘𝐵) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1352 | . . 3 ⊢ ⊤ | |
2 | fveq2 5494 | . . . 4 ⊢ (𝑥 = 𝑦 → (exp‘𝑥) = (exp‘𝑦)) | |
3 | fveq2 5494 | . . . 4 ⊢ (𝑥 = 𝐴 → (exp‘𝑥) = (exp‘𝐴)) | |
4 | fveq2 5494 | . . . 4 ⊢ (𝑥 = 𝐵 → (exp‘𝑥) = (exp‘𝐵)) | |
5 | ssid 3167 | . . . 4 ⊢ ℝ ⊆ ℝ | |
6 | reefcl 11620 | . . . . 5 ⊢ (𝑥 ∈ ℝ → (exp‘𝑥) ∈ ℝ) | |
7 | 6 | adantl 275 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (exp‘𝑥) ∈ ℝ) |
8 | efltim 11650 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → (exp‘𝑥) < (exp‘𝑦))) | |
9 | 8 | adantl 275 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 < 𝑦 → (exp‘𝑥) < (exp‘𝑦))) |
10 | 2, 3, 4, 5, 7, 9 | eqord1 8391 | . . 3 ⊢ ((⊤ ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → (𝐴 = 𝐵 ↔ (exp‘𝐴) = (exp‘𝐵))) |
11 | 1, 10 | mpan 422 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (exp‘𝐴) = (exp‘𝐵))) |
12 | 11 | bicomd 140 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((exp‘𝐴) = (exp‘𝐵) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ⊤wtru 1349 ∈ wcel 2141 class class class wbr 3987 ‘cfv 5196 ℝcr 7762 < clt 7943 expce 11594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 ax-arch 7882 ax-caucvg 7883 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-disj 3965 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-frec 6368 df-1o 6393 df-oadd 6397 df-er 6510 df-en 6716 df-dom 6717 df-fin 6718 df-sup 6958 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 df-inn 8868 df-2 8926 df-3 8927 df-4 8928 df-n0 9125 df-z 9202 df-uz 9477 df-q 9568 df-rp 9600 df-ico 9840 df-fz 9955 df-fzo 10088 df-seqfrec 10391 df-exp 10465 df-fac 10649 df-bc 10671 df-ihash 10699 df-cj 10795 df-re 10796 df-im 10797 df-rsqrt 10951 df-abs 10952 df-clim 11231 df-sumdc 11306 df-ef 11600 |
This theorem is referenced by: reeff1 11652 absefib 11722 efieq1re 11723 iooref1o 14028 |
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