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Mirrors > Home > MPE Home > Th. List > 5recm6rec | Structured version Visualization version 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 11575 | . . 3 ⊢ 5 ∈ ℂ | |
2 | 6cn 11578 | . . 3 ⊢ 6 ∈ ℂ | |
3 | 5re 11574 | . . . 4 ⊢ 5 ∈ ℝ | |
4 | 5pos 11596 | . . . 4 ⊢ 0 < 5 | |
5 | 3, 4 | gt0ne0ii 11026 | . . 3 ⊢ 5 ≠ 0 |
6 | 6re 11577 | . . . 4 ⊢ 6 ∈ ℝ | |
7 | 6pos 11597 | . . . 4 ⊢ 0 < 6 | |
8 | 6, 7 | gt0ne0ii 11026 | . . 3 ⊢ 6 ≠ 0 |
9 | 1, 2, 5, 8 | subreci 11320 | . 2 ⊢ ((1 / 5) − (1 / 6)) = ((6 − 5) / (5 · 6)) |
10 | ax-1cn 10444 | . . . 4 ⊢ 1 ∈ ℂ | |
11 | 5p1e6 11634 | . . . 4 ⊢ (5 + 1) = 6 | |
12 | 2, 1, 10, 11 | subaddrii 10825 | . . 3 ⊢ (6 − 5) = 1 |
13 | 6t5e30 12055 | . . . 4 ⊢ (6 · 5) = ;30 | |
14 | 2, 1, 13 | mulcomli 10499 | . . 3 ⊢ (5 · 6) = ;30 |
15 | 12, 14 | oveq12i 7031 | . 2 ⊢ ((6 − 5) / (5 · 6)) = (1 / ;30) |
16 | 9, 15 | eqtri 2818 | 1 ⊢ ((1 / 5) − (1 / 6)) = (1 / ;30) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 (class class class)co 7019 0cc0 10386 1c1 10387 · cmul 10391 − cmin 10719 / cdiv 11147 3c3 11543 5c5 11545 6c6 11546 ;cdc 11948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-7 11555 df-8 11556 df-9 11557 df-n0 11748 df-dec 11949 |
This theorem is referenced by: bpoly4 15246 |
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