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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 11213 | . . . 4 ⊢ 1 ∈ ℂ | |
| 2 | 8re 12362 | . . . . 5 ⊢ 8 ∈ ℝ | |
| 3 | 2 | recni 11275 | . . . 4 ⊢ 8 ∈ ℂ |
| 4 | 4cn 12351 | . . . 4 ⊢ 4 ∈ ℂ | |
| 5 | 3cn 12347 | . . . 4 ⊢ 3 ∈ ℂ | |
| 6 | 8pos 12378 | . . . . 5 ⊢ 0 < 8 | |
| 7 | 2, 6 | gt0ne0ii 11799 | . . . 4 ⊢ 8 ≠ 0 |
| 8 | 3ne0 12372 | . . . 4 ⊢ 3 ≠ 0 | |
| 9 | 1, 3, 4, 5, 7, 8 | divmuldivi 12027 | . . 3 ⊢ ((1 / 8) · (4 / 3)) = ((1 · 4) / (8 · 3)) |
| 10 | 1, 4 | mulcomi 11269 | . . . 4 ⊢ (1 · 4) = (4 · 1) |
| 11 | 2cn 12341 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
| 12 | 4, 11, 5 | mul32i 11457 | . . . . . . 7 ⊢ ((4 · 2) · 3) = ((4 · 3) · 2) |
| 13 | 4t2e8 12434 | . . . . . . . 8 ⊢ (4 · 2) = 8 | |
| 14 | 13 | oveq1i 7441 | . . . . . . 7 ⊢ ((4 · 2) · 3) = (8 · 3) |
| 15 | 12, 14 | eqtr3i 2767 | . . . . . 6 ⊢ ((4 · 3) · 2) = (8 · 3) |
| 16 | 4, 5, 11 | mulassi 11272 | . . . . . 6 ⊢ ((4 · 3) · 2) = (4 · (3 · 2)) |
| 17 | 15, 16 | eqtr3i 2767 | . . . . 5 ⊢ (8 · 3) = (4 · (3 · 2)) |
| 18 | 3t2e6 12432 | . . . . . 6 ⊢ (3 · 2) = 6 | |
| 19 | 18 | oveq2i 7442 | . . . . 5 ⊢ (4 · (3 · 2)) = (4 · 6) |
| 20 | 17, 19 | eqtri 2765 | . . . 4 ⊢ (8 · 3) = (4 · 6) |
| 21 | 10, 20 | oveq12i 7443 | . . 3 ⊢ ((1 · 4) / (8 · 3)) = ((4 · 1) / (4 · 6)) |
| 22 | 9, 21 | eqtri 2765 | . 2 ⊢ ((1 / 8) · (4 / 3)) = ((4 · 1) / (4 · 6)) |
| 23 | 6re 12356 | . . . 4 ⊢ 6 ∈ ℝ | |
| 24 | 23 | recni 11275 | . . 3 ⊢ 6 ∈ ℂ |
| 25 | 6pos 12376 | . . . 4 ⊢ 0 < 6 | |
| 26 | 23, 25 | gt0ne0ii 11799 | . . 3 ⊢ 6 ≠ 0 |
| 27 | 4ne0 12374 | . . 3 ⊢ 4 ≠ 0 | |
| 28 | divcan5 11969 | . . . 4 ⊢ ((1 ∈ ℂ ∧ (6 ∈ ℂ ∧ 6 ≠ 0) ∧ (4 ∈ ℂ ∧ 4 ≠ 0)) → ((4 · 1) / (4 · 6)) = (1 / 6)) | |
| 29 | 1, 28 | mp3an1 1450 | . . 3 ⊢ (((6 ∈ ℂ ∧ 6 ≠ 0) ∧ (4 ∈ ℂ ∧ 4 ≠ 0)) → ((4 · 1) / (4 · 6)) = (1 / 6)) |
| 30 | 24, 26, 4, 27, 29 | mp4an 693 | . 2 ⊢ ((4 · 1) / (4 · 6)) = (1 / 6) |
| 31 | 22, 30 | eqtri 2765 | 1 ⊢ ((1 / 8) · (4 / 3)) = (1 / 6) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 / cdiv 11920 2c2 12321 3c3 12322 4c4 12323 6c6 12325 8c8 12327 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 |
| This theorem is referenced by: (None) |
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