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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 14124 | . . . . . 6 ⊢ (𝑋 ∈ ℤ → (-1↑𝑋) ∈ ℤ) | |
2 | 1 | zcnd 12721 | . . . . 5 ⊢ (𝑋 ∈ ℤ → (-1↑𝑋) ∈ ℂ) |
3 | 2 | adantr 480 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑋) ∈ ℂ) |
4 | m1expcl 14124 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ ℤ) | |
5 | 4 | zcnd 12721 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ ℂ) |
6 | 5 | adantl 481 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ∈ ℂ) |
7 | neg1cn 12378 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | neg1ne0 12380 | . . . . . 6 ⊢ -1 ≠ 0 | |
9 | expne0i 14132 | . . . . . 6 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ≠ 0) | |
10 | 7, 8, 9 | mp3an12 1450 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ≠ 0) |
11 | 10 | adantl 481 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑𝑌) ≠ 0) |
12 | 3, 6, 11 | divrecd 12044 | . . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) / (-1↑𝑌)) = ((-1↑𝑋) · (1 / (-1↑𝑌)))) |
13 | m1expcl2 14123 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → (-1↑𝑌) ∈ {-1, 1}) | |
14 | elpri 4654 | . . . . . 6 ⊢ ((-1↑𝑌) ∈ {-1, 1} → ((-1↑𝑌) = -1 ∨ (-1↑𝑌) = 1)) | |
15 | ax-1cn 11211 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
16 | ax-1ne0 11222 | . . . . . . . . . 10 ⊢ 1 ≠ 0 | |
17 | divneg2 11989 | . . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → -(1 / 1) = (1 / -1)) | |
18 | 15, 15, 16, 17 | mp3an 1460 | . . . . . . . . 9 ⊢ -(1 / 1) = (1 / -1) |
19 | 1div1e1 11956 | . . . . . . . . . 10 ⊢ (1 / 1) = 1 | |
20 | 19 | negeqi 11499 | . . . . . . . . 9 ⊢ -(1 / 1) = -1 |
21 | 18, 20 | eqtr3i 2765 | . . . . . . . 8 ⊢ (1 / -1) = -1 |
22 | oveq2 7439 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (1 / -1)) | |
23 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = -1 → (-1↑𝑌) = -1) | |
24 | 21, 22, 23 | 3eqtr4a 2801 | . . . . . . 7 ⊢ ((-1↑𝑌) = -1 → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
25 | oveq2 7439 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (1 / (-1↑𝑌)) = (1 / 1)) | |
26 | id 22 | . . . . . . . 8 ⊢ ((-1↑𝑌) = 1 → (-1↑𝑌) = 1) | |
27 | 19, 25, 26 | 3eqtr4a 2801 | . . . . . . 7 ⊢ ((-1↑𝑌) = 1 → (1 / (-1↑𝑌)) = (-1↑𝑌)) |
28 | 24, 27 | jaoi 857 | . . . . . 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 7447 | . . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) · (1 / (-1↑𝑌))) = ((-1↑𝑋) · (-1↑𝑌))) |
32 | 12, 31 | eqtrd 2775 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → ((-1↑𝑋) / (-1↑𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) |
33 | expsub 14148 | . . 3 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) → (-1↑(𝑋 − 𝑌)) = ((-1↑𝑋) / (-1↑𝑌))) | |
34 | 7, 8, 33 | mpanl12 702 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = ((-1↑𝑋) / (-1↑𝑌))) |
35 | expaddz 14144 | . . 3 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) → (-1↑(𝑋 + 𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) | |
36 | 7, 8, 35 | mpanl12 702 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 + 𝑌)) = ((-1↑𝑋) · (-1↑𝑌))) |
37 | 32, 34, 36 | 3eqtr4d 2785 | 1 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = (-1↑(𝑋 + 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {cpr 4633 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 -cneg 11491 / cdiv 11918 ℤcz 12611 ↑cexp 14099 |
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-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 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 |
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-op 4638 df-uni 4913 df-iun 4998 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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 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-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-exp 14100 |
This theorem is referenced by: psgnuni 19532 41prothprmlem2 47543 |
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