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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 13455 | . . . . . 6 ⊢ (𝑋 ∈ ℤ → (-1↑𝑋) ∈ ℤ) | |
2 | 1 | zcnd 12091 | . . . . 5 ⊢ (𝑋 ∈ ℤ → (-1↑𝑋) ∈ ℂ) |
3 | 2 | adantr 483 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑋) ∈ ℂ) |
4 | m1expcl 13455 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ ℤ) | |
5 | 4 | zcnd 12091 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ ℂ) |
6 | 5 | adantl 484 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ∈ ℂ) |
7 | neg1cn 11754 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | neg1ne0 11756 | . . . . . 6 ⊢ -1 ≠ 0 | |
9 | expne0i 13464 | . . . . . 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 484 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ≠ 0) |
12 | 3, 6, 11 | divrecd 11421 | . . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) / (-1↑𝑌)) = ((-1↑𝑋) · (1 / (-1↑𝑌)))) |
13 | m1expcl2 13454 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ {-1, 1}) | |
14 | elpri 4591 | . . . . . 6 ⊢ ((-1↑𝑌) ∈ {-1, 1} → ((-1↑𝑌) = -1 ∨ (-1↑𝑌) = 1)) | |
15 | ax-1cn 10597 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
16 | ax-1ne0 10608 | . . . . . . . . . 10 ⊢ 1 ≠ 0 | |
17 | divneg2 11366 | . . . . . . . . . 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 11332 | . . . . . . . . . 10 ⊢ (1 / 1) = 1 | |
20 | 19 | negeqi 10881 | . . . . . . . . 9 ⊢ -(1 / 1) = -1 |
21 | 18, 20 | eqtr3i 2848 | . . . . . . . 8 ⊢ (1 / -1) = -1 |
22 | oveq2 7166 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (1 / -1)) | |
23 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (-1↑𝑌) = -1) | |
24 | 21, 22, 23 | 3eqtr4a 2884 | . . . . . . 7 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
25 | oveq2 7166 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (1 / (-1↑𝑌)) = (1 / 1)) | |
26 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (-1↑𝑌) = 1) | |
27 | 19, 25, 26 | 3eqtr4a 2884 | . . . . . . 7 ⊢ ((-1↑𝑌) = 1 → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
28 | 24, 27 | jaoi 853 | . . . . . 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 484 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
31 | 30 | oveq2d 7174 | . . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) · (1 / (-1↑𝑌))) = ((-1↑𝑋) · (-1↑𝑌))) |
32 | 12, 31 | eqtrd 2858 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) / (-1↑𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) |
33 | expsub 13480 | . . 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 13476 | . . 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 2868 | 1 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = (-1↑(𝑋 + 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {cpr 4571 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 − cmin 10872 -cneg 10873 / cdiv 11299 ℤcz 11984 ↑cexp 13432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-exp 13433 |
This theorem is referenced by: psgnuni 18629 41prothprmlem2 43790 |
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