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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 12745 | . . . . . 6 ⊢ (𝑋 ∈ ℤ → (-1↑𝑋) ∈ ℤ) | |
2 | 1 | zcnd 11359 | . . . . 5 ⊢ (𝑋 ∈ ℤ → (-1↑𝑋) ∈ ℂ) |
3 | 2 | adantr 480 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑋) ∈ ℂ) |
4 | m1expcl 12745 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ ℤ) | |
5 | 4 | zcnd 11359 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ ℂ) |
6 | 5 | adantl 481 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ∈ ℂ) |
7 | neg1cn 11001 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | neg1ne0 11003 | . . . . . 6 ⊢ -1 ≠ 0 | |
9 | expne0i 12754 | . . . . . 6 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ≠ 0) | |
10 | 7, 8, 9 | mp3an12 1406 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ≠ 0) |
11 | 10 | adantl 481 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ≠ 0) |
12 | 3, 6, 11 | divrecd 10683 | . . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) / (-1↑𝑌)) = ((-1↑𝑋) · (1 / (-1↑𝑌)))) |
13 | m1expcl2 12744 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ {-1, 1}) | |
14 | elpri 4145 | . . . . . 6 ⊢ ((-1↑𝑌) ∈ {-1, 1} → ((-1↑𝑌) = -1 ∨ (-1↑𝑌) = 1)) | |
15 | ax-1cn 9873 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
16 | ax-1ne0 9884 | . . . . . . . . . 10 ⊢ 1 ≠ 0 | |
17 | divneg2 10628 | . . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → -(1 / 1) = (1 / -1)) | |
18 | 15, 15, 16, 17 | mp3an 1416 | . . . . . . . . 9 ⊢ -(1 / 1) = (1 / -1) |
19 | 1div1e1 10596 | . . . . . . . . . 10 ⊢ (1 / 1) = 1 | |
20 | 19 | negeqi 10153 | . . . . . . . . 9 ⊢ -(1 / 1) = -1 |
21 | 18, 20 | eqtr3i 2634 | . . . . . . . 8 ⊢ (1 / -1) = -1 |
22 | oveq2 6557 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (1 / -1)) | |
23 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (-1↑𝑌) = -1) | |
24 | 21, 22, 23 | 3eqtr4a 2670 | . . . . . . 7 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
25 | oveq2 6557 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (1 / (-1↑𝑌)) = (1 / 1)) | |
26 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (-1↑𝑌) = 1) | |
27 | 19, 25, 26 | 3eqtr4a 2670 | . . . . . . 7 ⊢ ((-1↑𝑌) = 1 → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
28 | 24, 27 | jaoi 393 | . . . . . 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 481 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
31 | 30 | oveq2d 6565 | . . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) · (1 / (-1↑𝑌))) = ((-1↑𝑋) · (-1↑𝑌))) |
32 | 12, 31 | eqtrd 2644 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) / (-1↑𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) |
33 | expsub 12770 | . . 3 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) → (-1↑(𝑋 − 𝑌)) = ((-1↑𝑋) / (-1↑𝑌))) | |
34 | 7, 8, 33 | mpanl12 714 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = ((-1↑𝑋) / (-1↑𝑌))) |
35 | expaddz 12766 | . . 3 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) → (-1↑(𝑋 + 𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) | |
36 | 7, 8, 35 | mpanl12 714 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 + 𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) |
37 | 32, 34, 36 | 3eqtr4d 2654 | 1 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = (-1↑(𝑋 + 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {cpr 4127 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 / cdiv 10563 ℤcz 11254 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: psgnuni 17742 41prothprmlem2 40073 |
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