| Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > cxpi11d | Structured version Visualization version GIF version | ||
| Description: i to the powers of 𝐴 and 𝐵 are equal iff 𝐴 and 𝐵 are a multiple of 4 apart. EDITORIAL: This theorem may be revised to a more convenient form. (Contributed by SN, 25-Apr-2025.) |
| Ref | Expression |
|---|---|
| cxpi11d.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| cxpi11d.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| Ref | Expression |
|---|---|
| cxpi11d | ⊢ (𝜑 → ((i↑𝑐𝐴) = (i↑𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (4 · 𝑛)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-icn 11214 | . . . 4 ⊢ i ∈ ℂ | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → i ∈ ℂ) |
| 3 | cxpi11d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 4 | cxpi11d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 5 | ine0 11698 | . . . 4 ⊢ i ≠ 0 | |
| 6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → i ≠ 0) |
| 7 | ine1 42349 | . . . 4 ⊢ i ≠ 1 | |
| 8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → i ≠ 1) |
| 9 | 2, 3, 4, 6, 8 | cxp112d 42377 | . 2 ⊢ (𝜑 → ((i↑𝑐𝐴) = (i↑𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (((i · (2 · π)) · 𝑛) / (log‘i))))) |
| 10 | 2cn 12341 | . . . . . . . . . 10 ⊢ 2 ∈ ℂ | |
| 11 | picn 26501 | . . . . . . . . . 10 ⊢ π ∈ ℂ | |
| 12 | 10, 11 | mulcli 11268 | . . . . . . . . 9 ⊢ (2 · π) ∈ ℂ |
| 13 | 1, 12 | mulcli 11268 | . . . . . . . 8 ⊢ (i · (2 · π)) ∈ ℂ |
| 14 | 13 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (i · (2 · π)) ∈ ℂ) |
| 15 | zcn 12618 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
| 16 | logcl 26610 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ i ≠ 0) → (log‘i) ∈ ℂ) | |
| 17 | 1, 5, 16 | mp2an 692 | . . . . . . . 8 ⊢ (log‘i) ∈ ℂ |
| 18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (log‘i) ∈ ℂ) |
| 19 | logccne0 26620 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ i ≠ 1) → (log‘i) ≠ 0) | |
| 20 | 1, 5, 7, 19 | mp3an 1463 | . . . . . . . 8 ⊢ (log‘i) ≠ 0 |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (log‘i) ≠ 0) |
| 22 | 14, 15, 18, 21 | div23d 12080 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (((i · (2 · π)) · 𝑛) / (log‘i)) = (((i · (2 · π)) / (log‘i)) · 𝑛)) |
| 23 | logi 26629 | . . . . . . . . 9 ⊢ (log‘i) = (i · (π / 2)) | |
| 24 | 23 | oveq2i 7442 | . . . . . . . 8 ⊢ ((i · (2 · π)) / (log‘i)) = ((i · (2 · π)) / (i · (π / 2))) |
| 25 | 12 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → (2 · π) ∈ ℂ) |
| 26 | 2ne0 12370 | . . . . . . . . . . . 12 ⊢ 2 ≠ 0 | |
| 27 | 11, 10, 26 | divcli 12009 | . . . . . . . . . . 11 ⊢ (π / 2) ∈ ℂ |
| 28 | 27 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → (π / 2) ∈ ℂ) |
| 29 | 1 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → i ∈ ℂ) |
| 30 | pine0 42348 | . . . . . . . . . . . 12 ⊢ π ≠ 0 | |
| 31 | 11, 10, 30, 26 | divne0i 12015 | . . . . . . . . . . 11 ⊢ (π / 2) ≠ 0 |
| 32 | 31 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → (π / 2) ≠ 0) |
| 33 | 5 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → i ≠ 0) |
| 34 | 25, 28, 29, 32, 33 | divcan5d 12069 | . . . . . . . . 9 ⊢ (⊤ → ((i · (2 · π)) / (i · (π / 2))) = ((2 · π) / (π / 2))) |
| 35 | 34 | mptru 1547 | . . . . . . . 8 ⊢ ((i · (2 · π)) / (i · (π / 2))) = ((2 · π) / (π / 2)) |
| 36 | 10, 11, 27, 31 | divassi 12023 | . . . . . . . . 9 ⊢ ((2 · π) / (π / 2)) = (2 · (π / (π / 2))) |
| 37 | 11 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → π ∈ ℂ) |
| 38 | 2cnd 12344 | . . . . . . . . . . . 12 ⊢ (⊤ → 2 ∈ ℂ) | |
| 39 | 30 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → π ≠ 0) |
| 40 | 26 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → 2 ≠ 0) |
| 41 | 37, 38, 39, 40 | ddcand 12063 | . . . . . . . . . . 11 ⊢ (⊤ → (π / (π / 2)) = 2) |
| 42 | 41 | mptru 1547 | . . . . . . . . . 10 ⊢ (π / (π / 2)) = 2 |
| 43 | 42 | oveq2i 7442 | . . . . . . . . 9 ⊢ (2 · (π / (π / 2))) = (2 · 2) |
| 44 | 2t2e4 12430 | . . . . . . . . 9 ⊢ (2 · 2) = 4 | |
| 45 | 36, 43, 44 | 3eqtri 2769 | . . . . . . . 8 ⊢ ((2 · π) / (π / 2)) = 4 |
| 46 | 24, 35, 45 | 3eqtri 2769 | . . . . . . 7 ⊢ ((i · (2 · π)) / (log‘i)) = 4 |
| 47 | 46 | oveq1i 7441 | . . . . . 6 ⊢ (((i · (2 · π)) / (log‘i)) · 𝑛) = (4 · 𝑛) |
| 48 | 22, 47 | eqtrdi 2793 | . . . . 5 ⊢ (𝑛 ∈ ℤ → (((i · (2 · π)) · 𝑛) / (log‘i)) = (4 · 𝑛)) |
| 49 | 48 | oveq2d 7447 | . . . 4 ⊢ (𝑛 ∈ ℤ → (𝐵 + (((i · (2 · π)) · 𝑛) / (log‘i))) = (𝐵 + (4 · 𝑛))) |
| 50 | 49 | eqeq2d 2748 | . . 3 ⊢ (𝑛 ∈ ℤ → (𝐴 = (𝐵 + (((i · (2 · π)) · 𝑛) / (log‘i))) ↔ 𝐴 = (𝐵 + (4 · 𝑛)))) |
| 51 | 50 | rexbiia 3092 | . 2 ⊢ (∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (((i · (2 · π)) · 𝑛) / (log‘i))) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (4 · 𝑛))) |
| 52 | 9, 51 | bitrdi 287 | 1 ⊢ (𝜑 → ((i↑𝑐𝐴) = (i↑𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (4 · 𝑛)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 / cdiv 11920 2c2 12321 4c4 12323 ℤcz 12613 πcpi 16102 logclog 26596 ↑𝑐ccxp 26597 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |