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Mirrors > Home > MPE Home > Th. List > m1expaddsub | Structured version Visualization version GIF version |
Description: Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
Ref | Expression |
---|---|
m1expaddsub | ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = (-1↑(𝑋 + 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m1expcl 14101 | . . . . . 6 ⊢ (𝑋 ∈ ℤ → (-1↑𝑋) ∈ ℤ) | |
2 | 1 | zcnd 12714 | . . . . 5 ⊢ (𝑋 ∈ ℤ → (-1↑𝑋) ∈ ℂ) |
3 | 2 | adantr 479 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑋) ∈ ℂ) |
4 | m1expcl 14101 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ ℤ) | |
5 | 4 | zcnd 12714 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ ℂ) |
6 | 5 | adantl 480 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ∈ ℂ) |
7 | neg1cn 12373 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | neg1ne0 12375 | . . . . . 6 ⊢ -1 ≠ 0 | |
9 | expne0i 14109 | . . . . . 6 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ≠ 0) | |
10 | 7, 8, 9 | mp3an12 1447 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ≠ 0) |
11 | 10 | adantl 480 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ≠ 0) |
12 | 3, 6, 11 | divrecd 12040 | . . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) / (-1↑𝑌)) = ((-1↑𝑋) · (1 / (-1↑𝑌)))) |
13 | m1expcl2 14100 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ {-1, 1}) | |
14 | elpri 4655 | . . . . . 6 ⊢ ((-1↑𝑌) ∈ {-1, 1} → ((-1↑𝑌) = -1 ∨ (-1↑𝑌) = 1)) | |
15 | ax-1cn 11212 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
16 | ax-1ne0 11223 | . . . . . . . . . 10 ⊢ 1 ≠ 0 | |
17 | divneg2 11985 | . . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → -(1 / 1) = (1 / -1)) | |
18 | 15, 15, 16, 17 | mp3an 1457 | . . . . . . . . 9 ⊢ -(1 / 1) = (1 / -1) |
19 | 1div1e1 11951 | . . . . . . . . . 10 ⊢ (1 / 1) = 1 | |
20 | 19 | negeqi 11499 | . . . . . . . . 9 ⊢ -(1 / 1) = -1 |
21 | 18, 20 | eqtr3i 2755 | . . . . . . . 8 ⊢ (1 / -1) = -1 |
22 | oveq2 7431 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (1 / -1)) | |
23 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (-1↑𝑌) = -1) | |
24 | 21, 22, 23 | 3eqtr4a 2791 | . . . . . . 7 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
25 | oveq2 7431 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (1 / (-1↑𝑌)) = (1 / 1)) | |
26 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (-1↑𝑌) = 1) | |
27 | 19, 25, 26 | 3eqtr4a 2791 | . . . . . . 7 ⊢ ((-1↑𝑌) = 1 → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
28 | 24, 27 | jaoi 855 | . . . . . 6 ⊢ (((-1↑𝑌) = -1 ∨ (-1↑𝑌) = 1) → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
29 | 13, 14, 28 | 3syl 18 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
30 | 29 | adantl 480 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
31 | 30 | oveq2d 7439 | . . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) · (1 / (-1↑𝑌))) = ((-1↑𝑋) · (-1↑𝑌))) |
32 | 12, 31 | eqtrd 2765 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) / (-1↑𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) |
33 | expsub 14125 | . . 3 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) → (-1↑(𝑋 − 𝑌)) = ((-1↑𝑋) / (-1↑𝑌))) | |
34 | 7, 8, 33 | mpanl12 700 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = ((-1↑𝑋) / (-1↑𝑌))) |
35 | expaddz 14121 | . . 3 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) → (-1↑(𝑋 + 𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) | |
36 | 7, 8, 35 | mpanl12 700 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 + 𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) |
37 | 32, 34, 36 | 3eqtr4d 2775 | 1 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = (-1↑(𝑋 + 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 {cpr 4634 (class class class)co 7423 ℂcc 11152 0cc0 11154 1c1 11155 + caddc 11157 · cmul 11159 − cmin 11490 -cneg 11491 / cdiv 11917 ℤcz 12605 ↑cexp 14076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-n0 12520 df-z 12606 df-uz 12870 df-seq 14017 df-exp 14077 |
This theorem is referenced by: psgnuni 19492 41prothprmlem2 47127 |
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