Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11422 | . . 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 4592 | . . . . . 6 ⊢ ((-1↑𝑌) ∈ {-1, 1} → ((-1↑𝑌) = -1 ∨ (-1↑𝑌) = 1)) | |
15 | ax-1cn 10598 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
16 | ax-1ne0 10609 | . . . . . . . . . 10 ⊢ 1 ≠ 0 | |
17 | divneg2 11367 | . . . . . . . . . 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 11333 | . . . . . . . . . 10 ⊢ (1 / 1) = 1 | |
20 | 19 | negeqi 10882 | . . . . . . . . 9 ⊢ -(1 / 1) = -1 |
21 | 18, 20 | eqtr3i 2849 | . . . . . . . 8 ⊢ (1 / -1) = -1 |
22 | oveq2 7167 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (1 / -1)) | |
23 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (-1↑𝑌) = -1) | |
24 | 21, 22, 23 | 3eqtr4a 2885 | . . . . . . 7 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
25 | oveq2 7167 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (1 / (-1↑𝑌)) = (1 / 1)) | |
26 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (-1↑𝑌) = 1) | |
27 | 19, 25, 26 | 3eqtr4a 2885 | . . . . . . 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 7175 | . . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) · (1 / (-1↑𝑌))) = ((-1↑𝑋) · (-1↑𝑌))) |
32 | 12, 31 | eqtrd 2859 | . 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 2869 | 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 1536 ∈ wcel 2113 ≠ wne 3019 {cpr 4572 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 / cdiv 11300 ℤcz 11984 ↑cexp 13432 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-exp 13433 |
This theorem is referenced by: psgnuni 18630 41prothprmlem2 43790 |
Copyright terms: Public domain | W3C validator |