![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > halfpm6th | GIF version |
Description: One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.) |
Ref | Expression |
---|---|
halfpm6th | ⊢ (((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3cn 7770 | . . . . . 6 ⊢ 3 ∈ ℂ | |
2 | ax-1cn 6776 | . . . . . 6 ⊢ 1 ∈ ℂ | |
3 | 2cn 7766 | . . . . . 6 ⊢ 2 ∈ ℂ | |
4 | 3re 7769 | . . . . . . 7 ⊢ 3 ∈ ℝ | |
5 | 3pos 7790 | . . . . . . 7 ⊢ 0 < 3 | |
6 | 4, 5 | gt0ap0ii 7410 | . . . . . 6 ⊢ 3 # 0 |
7 | 2ap0 7789 | . . . . . 6 ⊢ 2 # 0 | |
8 | 1, 1, 2, 3, 6, 7 | divmuldivapi 7530 | . . . . 5 ⊢ ((3 / 3) · (1 / 2)) = ((3 · 1) / (3 · 2)) |
9 | 1, 6 | dividapi 7503 | . . . . . . 7 ⊢ (3 / 3) = 1 |
10 | 9 | oveq1i 5465 | . . . . . 6 ⊢ ((3 / 3) · (1 / 2)) = (1 · (1 / 2)) |
11 | halfcn 7917 | . . . . . . 7 ⊢ (1 / 2) ∈ ℂ | |
12 | 11 | mulid2i 6828 | . . . . . 6 ⊢ (1 · (1 / 2)) = (1 / 2) |
13 | 10, 12 | eqtri 2057 | . . . . 5 ⊢ ((3 / 3) · (1 / 2)) = (1 / 2) |
14 | 1 | mulid1i 6827 | . . . . . 6 ⊢ (3 · 1) = 3 |
15 | 3t2e6 7849 | . . . . . 6 ⊢ (3 · 2) = 6 | |
16 | 14, 15 | oveq12i 5467 | . . . . 5 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6) |
17 | 8, 13, 16 | 3eqtr3i 2065 | . . . 4 ⊢ (1 / 2) = (3 / 6) |
18 | 17 | oveq1i 5465 | . . 3 ⊢ ((1 / 2) − (1 / 6)) = ((3 / 6) − (1 / 6)) |
19 | 6cn 7777 | . . . . 5 ⊢ 6 ∈ ℂ | |
20 | 6re 7776 | . . . . . 6 ⊢ 6 ∈ ℝ | |
21 | 6pos 7795 | . . . . . 6 ⊢ 0 < 6 | |
22 | 20, 21 | gt0ap0ii 7410 | . . . . 5 ⊢ 6 # 0 |
23 | 19, 22 | pm3.2i 257 | . . . 4 ⊢ (6 ∈ ℂ ∧ 6 # 0) |
24 | divsubdirap 7466 | . . . 4 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ ∧ (6 ∈ ℂ ∧ 6 # 0)) → ((3 − 1) / 6) = ((3 / 6) − (1 / 6))) | |
25 | 1, 2, 23, 24 | mp3an 1231 | . . 3 ⊢ ((3 − 1) / 6) = ((3 / 6) − (1 / 6)) |
26 | 3m1e2 7814 | . . . . 5 ⊢ (3 − 1) = 2 | |
27 | 26 | oveq1i 5465 | . . . 4 ⊢ ((3 − 1) / 6) = (2 / 6) |
28 | 3 | mulid2i 6828 | . . . . 5 ⊢ (1 · 2) = 2 |
29 | 28, 15 | oveq12i 5467 | . . . 4 ⊢ ((1 · 2) / (3 · 2)) = (2 / 6) |
30 | 3, 7 | dividapi 7503 | . . . . . 6 ⊢ (2 / 2) = 1 |
31 | 30 | oveq2i 5466 | . . . . 5 ⊢ ((1 / 3) · (2 / 2)) = ((1 / 3) · 1) |
32 | 2, 1, 3, 3, 6, 7 | divmuldivapi 7530 | . . . . 5 ⊢ ((1 / 3) · (2 / 2)) = ((1 · 2) / (3 · 2)) |
33 | 1, 6 | recclapi 7500 | . . . . . 6 ⊢ (1 / 3) ∈ ℂ |
34 | 33 | mulid1i 6827 | . . . . 5 ⊢ ((1 / 3) · 1) = (1 / 3) |
35 | 31, 32, 34 | 3eqtr3i 2065 | . . . 4 ⊢ ((1 · 2) / (3 · 2)) = (1 / 3) |
36 | 27, 29, 35 | 3eqtr2i 2063 | . . 3 ⊢ ((3 − 1) / 6) = (1 / 3) |
37 | 18, 25, 36 | 3eqtr2i 2063 | . 2 ⊢ ((1 / 2) − (1 / 6)) = (1 / 3) |
38 | 1, 2, 19, 22 | divdirapi 7527 | . . . 4 ⊢ ((3 + 1) / 6) = ((3 / 6) + (1 / 6)) |
39 | df-4 7755 | . . . . 5 ⊢ 4 = (3 + 1) | |
40 | 39 | oveq1i 5465 | . . . 4 ⊢ (4 / 6) = ((3 + 1) / 6) |
41 | 17 | oveq1i 5465 | . . . 4 ⊢ ((1 / 2) + (1 / 6)) = ((3 / 6) + (1 / 6)) |
42 | 38, 40, 41 | 3eqtr4ri 2068 | . . 3 ⊢ ((1 / 2) + (1 / 6)) = (4 / 6) |
43 | 2t2e4 7847 | . . . 4 ⊢ (2 · 2) = 4 | |
44 | 43, 15 | oveq12i 5467 | . . 3 ⊢ ((2 · 2) / (3 · 2)) = (4 / 6) |
45 | 30 | oveq2i 5466 | . . . 4 ⊢ ((2 / 3) · (2 / 2)) = ((2 / 3) · 1) |
46 | 3, 1, 3, 3, 6, 7 | divmuldivapi 7530 | . . . 4 ⊢ ((2 / 3) · (2 / 2)) = ((2 · 2) / (3 · 2)) |
47 | 3, 1, 6 | divclapi 7512 | . . . . 5 ⊢ (2 / 3) ∈ ℂ |
48 | 47 | mulid1i 6827 | . . . 4 ⊢ ((2 / 3) · 1) = (2 / 3) |
49 | 45, 46, 48 | 3eqtr3i 2065 | . . 3 ⊢ ((2 · 2) / (3 · 2)) = (2 / 3) |
50 | 42, 44, 49 | 3eqtr2i 2063 | . 2 ⊢ ((1 / 2) + (1 / 6)) = (2 / 3) |
51 | 37, 50 | pm3.2i 257 | 1 ⊢ (((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∈ wcel 1390 class class class wbr 3755 (class class class)co 5455 ℂcc 6709 0cc0 6711 1c1 6712 + caddc 6714 · cmul 6716 − cmin 6979 # cap 7365 / cdiv 7433 2c2 7744 3c3 7745 4c4 7746 6c6 7748 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-mulrcl 6782 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-precex 6793 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 ax-pre-apti 6798 ax-pre-ltadd 6799 ax-pre-mulgt0 6800 ax-pre-mulext 6801 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rmo 2308 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 df-reap 7359 df-ap 7366 df-div 7434 df-2 7753 df-3 7754 df-4 7755 df-5 7756 df-6 7757 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |