![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > 5recm6rec | GIF version |
Description: One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.) |
Ref | Expression |
---|---|
5recm6rec | ⊢ ((1 / 5) − (1 / 6)) = (1 / ;30) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5cn 8975 | . . 3 ⊢ 5 ∈ ℂ | |
2 | 6cn 8977 | . . 3 ⊢ 6 ∈ ℂ | |
3 | 5re 8974 | . . . 4 ⊢ 5 ∈ ℝ | |
4 | 5pos 8995 | . . . 4 ⊢ 0 < 5 | |
5 | 3, 4 | gt0ap0ii 8562 | . . 3 ⊢ 5 # 0 |
6 | 6re 8976 | . . . 4 ⊢ 6 ∈ ℝ | |
7 | 6pos 8996 | . . . 4 ⊢ 0 < 6 | |
8 | 6, 7 | gt0ap0ii 8562 | . . 3 ⊢ 6 # 0 |
9 | 1, 2, 5, 8 | subrecapi 8773 | . 2 ⊢ ((1 / 5) − (1 / 6)) = ((6 − 5) / (5 · 6)) |
10 | ax-1cn 7882 | . . . 4 ⊢ 1 ∈ ℂ | |
11 | 5p1e6 9032 | . . . 4 ⊢ (5 + 1) = 6 | |
12 | 2, 1, 10, 11 | subaddrii 8223 | . . 3 ⊢ (6 − 5) = 1 |
13 | 6t5e30 9466 | . . . 4 ⊢ (6 · 5) = ;30 | |
14 | 2, 1, 13 | mulcomli 7942 | . . 3 ⊢ (5 · 6) = ;30 |
15 | 12, 14 | oveq12i 5880 | . 2 ⊢ ((6 − 5) / (5 · 6)) = (1 / ;30) |
16 | 9, 15 | eqtri 2198 | 1 ⊢ ((1 / 5) − (1 / 6)) = (1 / ;30) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 (class class class)co 5868 0cc0 7789 1c1 7790 · cmul 7794 − cmin 8105 / cdiv 8605 3c3 8947 5c5 8949 6c6 8950 ;cdc 9360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-id 4289 df-po 4292 df-iso 4293 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-5 8957 df-6 8958 df-7 8959 df-8 8960 df-9 8961 df-n0 9153 df-dec 9361 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |