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| 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 9115 | . . 3 ⊢ 5 ∈ ℂ | |
| 2 | 6cn 9117 | . . 3 ⊢ 6 ∈ ℂ | |
| 3 | 5re 9114 | . . . 4 ⊢ 5 ∈ ℝ | |
| 4 | 5pos 9135 | . . . 4 ⊢ 0 < 5 | |
| 5 | 3, 4 | gt0ap0ii 8700 | . . 3 ⊢ 5 # 0 |
| 6 | 6re 9116 | . . . 4 ⊢ 6 ∈ ℝ | |
| 7 | 6pos 9136 | . . . 4 ⊢ 0 < 6 | |
| 8 | 6, 7 | gt0ap0ii 8700 | . . 3 ⊢ 6 # 0 |
| 9 | 1, 2, 5, 8 | subrecapi 8912 | . 2 ⊢ ((1 / 5) − (1 / 6)) = ((6 − 5) / (5 · 6)) |
| 10 | ax-1cn 8017 | . . . 4 ⊢ 1 ∈ ℂ | |
| 11 | 5p1e6 9173 | . . . 4 ⊢ (5 + 1) = 6 | |
| 12 | 2, 1, 10, 11 | subaddrii 8360 | . . 3 ⊢ (6 − 5) = 1 |
| 13 | 6t5e30 9609 | . . . 4 ⊢ (6 · 5) = ;30 | |
| 14 | 2, 1, 13 | mulcomli 8078 | . . 3 ⊢ (5 · 6) = ;30 |
| 15 | 12, 14 | oveq12i 5955 | . 2 ⊢ ((6 − 5) / (5 · 6)) = (1 / ;30) |
| 16 | 9, 15 | eqtri 2225 | 1 ⊢ ((1 / 5) − (1 / 6)) = (1 / ;30) |
| Colors of variables: wff set class |
| Syntax hints: = wceq 1372 (class class class)co 5943 0cc0 7924 1c1 7925 · cmul 7929 − cmin 8242 / cdiv 8744 3c3 9087 5c5 9089 6c6 9090 ;cdc 9503 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-br 4044 df-opab 4105 df-id 4339 df-po 4342 df-iso 4343 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-iota 5231 df-fun 5272 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 df-div 8745 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-5 9097 df-6 9098 df-7 9099 df-8 9100 df-9 9101 df-n0 9295 df-dec 9504 |
| This theorem is referenced by: (None) |
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