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Mirrors > Home > MPE Home > Th. List > 8th4div3 | Structured version Visualization version GIF version |
Description: An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.) |
Ref | Expression |
---|---|
8th4div3 | ⊢ ((1 / 8) · (4 / 3)) = (1 / 6) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10283 | . . . 4 ⊢ 1 ∈ ℂ | |
2 | 8re 11413 | . . . . 5 ⊢ 8 ∈ ℝ | |
3 | 2 | recni 10344 | . . . 4 ⊢ 8 ∈ ℂ |
4 | 4cn 11398 | . . . 4 ⊢ 4 ∈ ℂ | |
5 | 3cn 11393 | . . . 4 ⊢ 3 ∈ ℂ | |
6 | 8pos 11431 | . . . . 5 ⊢ 0 < 8 | |
7 | 2, 6 | gt0ne0ii 10857 | . . . 4 ⊢ 8 ≠ 0 |
8 | 3ne0 11425 | . . . 4 ⊢ 3 ≠ 0 | |
9 | 1, 3, 4, 5, 7, 8 | divmuldivi 11078 | . . 3 ⊢ ((1 / 8) · (4 / 3)) = ((1 · 4) / (8 · 3)) |
10 | 1, 4 | mulcomi 10338 | . . . 4 ⊢ (1 · 4) = (4 · 1) |
11 | 2cn 11387 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
12 | 4, 11, 5 | mul32i 10523 | . . . . . . 7 ⊢ ((4 · 2) · 3) = ((4 · 3) · 2) |
13 | 4t2e8 11487 | . . . . . . . 8 ⊢ (4 · 2) = 8 | |
14 | 13 | oveq1i 6889 | . . . . . . 7 ⊢ ((4 · 2) · 3) = (8 · 3) |
15 | 12, 14 | eqtr3i 2824 | . . . . . 6 ⊢ ((4 · 3) · 2) = (8 · 3) |
16 | 4, 5, 11 | mulassi 10341 | . . . . . 6 ⊢ ((4 · 3) · 2) = (4 · (3 · 2)) |
17 | 15, 16 | eqtr3i 2824 | . . . . 5 ⊢ (8 · 3) = (4 · (3 · 2)) |
18 | 3t2e6 11485 | . . . . . 6 ⊢ (3 · 2) = 6 | |
19 | 18 | oveq2i 6890 | . . . . 5 ⊢ (4 · (3 · 2)) = (4 · 6) |
20 | 17, 19 | eqtri 2822 | . . . 4 ⊢ (8 · 3) = (4 · 6) |
21 | 10, 20 | oveq12i 6891 | . . 3 ⊢ ((1 · 4) / (8 · 3)) = ((4 · 1) / (4 · 6)) |
22 | 9, 21 | eqtri 2822 | . 2 ⊢ ((1 / 8) · (4 / 3)) = ((4 · 1) / (4 · 6)) |
23 | 6re 11405 | . . . 4 ⊢ 6 ∈ ℝ | |
24 | 23 | recni 10344 | . . 3 ⊢ 6 ∈ ℂ |
25 | 6pos 11429 | . . . 4 ⊢ 0 < 6 | |
26 | 23, 25 | gt0ne0ii 10857 | . . 3 ⊢ 6 ≠ 0 |
27 | 4ne0 11427 | . . 3 ⊢ 4 ≠ 0 | |
28 | divcan5 11020 | . . . 4 ⊢ ((1 ∈ ℂ ∧ (6 ∈ ℂ ∧ 6 ≠ 0) ∧ (4 ∈ ℂ ∧ 4 ≠ 0)) → ((4 · 1) / (4 · 6)) = (1 / 6)) | |
29 | 1, 28 | mp3an1 1573 | . . 3 ⊢ (((6 ∈ ℂ ∧ 6 ≠ 0) ∧ (4 ∈ ℂ ∧ 4 ≠ 0)) → ((4 · 1) / (4 · 6)) = (1 / 6)) |
30 | 24, 26, 4, 27, 29 | mp4an 685 | . 2 ⊢ ((4 · 1) / (4 · 6)) = (1 / 6) |
31 | 22, 30 | eqtri 2822 | 1 ⊢ ((1 / 8) · (4 / 3)) = (1 / 6) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2972 (class class class)co 6879 ℂcc 10223 0cc0 10225 1c1 10226 · cmul 10230 / cdiv 10977 2c2 11367 3c3 11368 4c4 11369 6c6 11371 8c8 11373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-po 5234 df-so 5235 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-div 10978 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 |
This theorem is referenced by: (None) |
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