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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > threehalves | Structured version Visualization version GIF version |
Description: Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
Ref | Expression |
---|---|
threehalves | ⊢ (3 / 2) = (1.5) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3re 11705 | . . . . 5 ⊢ 3 ∈ ℝ | |
2 | 2re 11699 | . . . . 5 ⊢ 2 ∈ ℝ | |
3 | 2ne0 11729 | . . . . 5 ⊢ 2 ≠ 0 | |
4 | 1, 2, 3 | redivcli 11396 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
5 | 4 | recni 10644 | . . 3 ⊢ (3 / 2) ∈ ℂ |
6 | 1nn0 11901 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
7 | 5re 11712 | . . . . 5 ⊢ 5 ∈ ℝ | |
8 | dpcl 30593 | . . . . 5 ⊢ ((1 ∈ ℕ0 ∧ 5 ∈ ℝ) → (1.5) ∈ ℝ) | |
9 | 6, 7, 8 | mp2an 691 | . . . 4 ⊢ (1.5) ∈ ℝ |
10 | 9 | recni 10644 | . . 3 ⊢ (1.5) ∈ ℂ |
11 | 2cnne0 11835 | . . 3 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
12 | 5, 10, 11 | 3pm3.2i 1336 | . 2 ⊢ ((3 / 2) ∈ ℂ ∧ (1.5) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) |
13 | 5nn0 11905 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
14 | 3nn0 11903 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
15 | 0nn0 11900 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | eqid 2798 | . . . . . 6 ⊢ ;15 = ;15 | |
17 | df-2 11688 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
18 | 17 | oveq1i 7145 | . . . . . . 7 ⊢ (2 + 1) = ((1 + 1) + 1) |
19 | 2p1e3 11767 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
20 | 18, 19 | eqtr3i 2823 | . . . . . 6 ⊢ ((1 + 1) + 1) = 3 |
21 | 5p5e10 12157 | . . . . . 6 ⊢ (5 + 5) = ;10 | |
22 | 6, 13, 6, 13, 16, 16, 20, 15, 21 | decaddc 12141 | . . . . 5 ⊢ (;15 + ;15) = ;30 |
23 | 6, 13, 6, 13, 14, 15, 22 | dpadd 30613 | . . . 4 ⊢ ((1.5) + (1.5)) = (3.0) |
24 | 14 | dp0u 30603 | . . . 4 ⊢ (3.0) = 3 |
25 | 23, 24 | eqtri 2821 | . . 3 ⊢ ((1.5) + (1.5)) = 3 |
26 | 10 | times2i 11764 | . . 3 ⊢ ((1.5) · 2) = ((1.5) + (1.5)) |
27 | 1 | recni 10644 | . . . 4 ⊢ 3 ∈ ℂ |
28 | 11 | simpli 487 | . . . 4 ⊢ 2 ∈ ℂ |
29 | 27, 28, 3 | divcan1i 11373 | . . 3 ⊢ ((3 / 2) · 2) = 3 |
30 | 25, 26, 29 | 3eqtr4ri 2832 | . 2 ⊢ ((3 / 2) · 2) = ((1.5) · 2) |
31 | mulcan2 11267 | . . 3 ⊢ (((3 / 2) ∈ ℂ ∧ (1.5) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((3 / 2) · 2) = ((1.5) · 2) ↔ (3 / 2) = (1.5))) | |
32 | 31 | biimpa 480 | . 2 ⊢ ((((3 / 2) ∈ ℂ ∧ (1.5) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) ∧ ((3 / 2) · 2) = ((1.5) · 2)) → (3 / 2) = (1.5)) |
33 | 12, 30, 32 | mp2an 691 | 1 ⊢ (3 / 2) = (1.5) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 / cdiv 11286 2c2 11680 3c3 11681 5c5 11683 ℕ0cn0 11885 ;cdc 12086 .cdp 30590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-dec 12087 df-dp2 30574 df-dp 30591 |
This theorem is referenced by: (None) |
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