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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 11212 | . . . 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 11696 | . . . 4 ⊢ i ≠ 0 | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → i ≠ 0) |
7 | ine1 42328 | . . . 4 ⊢ i ≠ 1 | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → i ≠ 1) |
9 | 2, 3, 4, 6, 8 | cxp112d 42356 | . 2 ⊢ (𝜑 → ((i↑𝑐𝐴) = (i↑𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (((i · (2 · π)) · 𝑛) / (log‘i))))) |
10 | 2cn 12339 | . . . . . . . . . 10 ⊢ 2 ∈ ℂ | |
11 | picn 26516 | . . . . . . . . . 10 ⊢ π ∈ ℂ | |
12 | 10, 11 | mulcli 11266 | . . . . . . . . 9 ⊢ (2 · π) ∈ ℂ |
13 | 1, 12 | mulcli 11266 | . . . . . . . 8 ⊢ (i · (2 · π)) ∈ ℂ |
14 | 13 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (i · (2 · π)) ∈ ℂ) |
15 | zcn 12616 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
16 | logcl 26625 | . . . . . . . . 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 26635 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ i ≠ 1) → (log‘i) ≠ 0) | |
20 | 1, 5, 7, 19 | mp3an 1460 | . . . . . . . 8 ⊢ (log‘i) ≠ 0 |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (log‘i) ≠ 0) |
22 | 14, 15, 18, 21 | div23d 12078 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (((i · (2 · π)) · 𝑛) / (log‘i)) = (((i · (2 · π)) / (log‘i)) · 𝑛)) |
23 | logi 26644 | . . . . . . . . 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 12368 | . . . . . . . . . . . 12 ⊢ 2 ≠ 0 | |
27 | 11, 10, 26 | divcli 12007 | . . . . . . . . . . 11 ⊢ (π / 2) ∈ ℂ |
28 | 27 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → (π / 2) ∈ ℂ) |
29 | 1 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → i ∈ ℂ) |
30 | pine0 42327 | . . . . . . . . . . . 12 ⊢ π ≠ 0 | |
31 | 11, 10, 30, 26 | divne0i 12013 | . . . . . . . . . . 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 12067 | . . . . . . . . 9 ⊢ (⊤ → ((i · (2 · π)) / (i · (π / 2))) = ((2 · π) / (π / 2))) |
35 | 34 | mptru 1544 | . . . . . . . 8 ⊢ ((i · (2 · π)) / (i · (π / 2))) = ((2 · π) / (π / 2)) |
36 | 10, 11, 27, 31 | divassi 12021 | . . . . . . . . 9 ⊢ ((2 · π) / (π / 2)) = (2 · (π / (π / 2))) |
37 | 11 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → π ∈ ℂ) |
38 | 2cnd 12342 | . . . . . . . . . . . 12 ⊢ (⊤ → 2 ∈ ℂ) | |
39 | 30 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → π ≠ 0) |
40 | 26 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → 2 ≠ 0) |
41 | 37, 38, 39, 40 | ddcand 12061 | . . . . . . . . . . 11 ⊢ (⊤ → (π / (π / 2)) = 2) |
42 | 41 | mptru 1544 | . . . . . . . . . 10 ⊢ (π / (π / 2)) = 2 |
43 | 42 | oveq2i 7442 | . . . . . . . . 9 ⊢ (2 · (π / (π / 2))) = (2 · 2) |
44 | 2t2e4 12428 | . . . . . . . . 9 ⊢ (2 · 2) = 4 | |
45 | 36, 43, 44 | 3eqtri 2767 | . . . . . . . 8 ⊢ ((2 · π) / (π / 2)) = 4 |
46 | 24, 35, 45 | 3eqtri 2767 | . . . . . . 7 ⊢ ((i · (2 · π)) / (log‘i)) = 4 |
47 | 46 | oveq1i 7441 | . . . . . 6 ⊢ (((i · (2 · π)) / (log‘i)) · 𝑛) = (4 · 𝑛) |
48 | 22, 47 | eqtrdi 2791 | . . . . 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 2746 | . . 3 ⊢ (𝑛 ∈ ℤ → (𝐴 = (𝐵 + (((i · (2 · π)) · 𝑛) / (log‘i))) ↔ 𝐴 = (𝐵 + (4 · 𝑛)))) |
51 | 50 | rexbiia 3090 | . 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 1537 ⊤wtru 1538 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 ici 11155 + caddc 11156 · cmul 11158 / cdiv 11918 2c2 12319 4c4 12321 ℤcz 12611 πcpi 16099 logclog 26611 ↑𝑐ccxp 26612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614 |
This theorem is referenced by: (None) |
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