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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 12238 | . . . . 5 ⊢ 3 ∈ ℝ | |
2 | 2re 12232 | . . . . 5 ⊢ 2 ∈ ℝ | |
3 | 2ne0 12262 | . . . . 5 ⊢ 2 ≠ 0 | |
4 | 1, 2, 3 | redivcli 11927 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
5 | 4 | recni 11174 | . . 3 ⊢ (3 / 2) ∈ ℂ |
6 | 1nn0 12434 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
7 | 5re 12245 | . . . . 5 ⊢ 5 ∈ ℝ | |
8 | dpcl 31796 | . . . . 5 ⊢ ((1 ∈ ℕ0 ∧ 5 ∈ ℝ) → (1.5) ∈ ℝ) | |
9 | 6, 7, 8 | mp2an 691 | . . . 4 ⊢ (1.5) ∈ ℝ |
10 | 9 | recni 11174 | . . 3 ⊢ (1.5) ∈ ℂ |
11 | 2cnne0 12368 | . . 3 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
12 | 5, 10, 11 | 3pm3.2i 1340 | . 2 ⊢ ((3 / 2) ∈ ℂ ∧ (1.5) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) |
13 | 5nn0 12438 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
14 | 3nn0 12436 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
15 | 0nn0 12433 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | eqid 2733 | . . . . . 6 ⊢ ;15 = ;15 | |
17 | df-2 12221 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
18 | 17 | oveq1i 7368 | . . . . . . 7 ⊢ (2 + 1) = ((1 + 1) + 1) |
19 | 2p1e3 12300 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
20 | 18, 19 | eqtr3i 2763 | . . . . . 6 ⊢ ((1 + 1) + 1) = 3 |
21 | 5p5e10 12694 | . . . . . 6 ⊢ (5 + 5) = ;10 | |
22 | 6, 13, 6, 13, 16, 16, 20, 15, 21 | decaddc 12678 | . . . . 5 ⊢ (;15 + ;15) = ;30 |
23 | 6, 13, 6, 13, 14, 15, 22 | dpadd 31816 | . . . 4 ⊢ ((1.5) + (1.5)) = (3.0) |
24 | 14 | dp0u 31806 | . . . 4 ⊢ (3.0) = 3 |
25 | 23, 24 | eqtri 2761 | . . 3 ⊢ ((1.5) + (1.5)) = 3 |
26 | 10 | times2i 12297 | . . 3 ⊢ ((1.5) · 2) = ((1.5) + (1.5)) |
27 | 1 | recni 11174 | . . . 4 ⊢ 3 ∈ ℂ |
28 | 11 | simpli 485 | . . . 4 ⊢ 2 ∈ ℂ |
29 | 27, 28, 3 | divcan1i 11904 | . . 3 ⊢ ((3 / 2) · 2) = 3 |
30 | 25, 26, 29 | 3eqtr4ri 2772 | . 2 ⊢ ((3 / 2) · 2) = ((1.5) · 2) |
31 | mulcan2 11798 | . . 3 ⊢ (((3 / 2) ∈ ℂ ∧ (1.5) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((3 / 2) · 2) = ((1.5) · 2) ↔ (3 / 2) = (1.5))) | |
32 | 31 | biimpa 478 | . 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 (class class class)co 7358 ℂcc 11054 ℝcr 11055 0cc0 11056 1c1 11057 + caddc 11059 · cmul 11061 / cdiv 11817 2c2 12213 3c3 12214 5c5 12216 ℕ0cn0 12418 ;cdc 12623 .cdp 31793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-dec 12624 df-dp2 31777 df-dp 31794 |
This theorem is referenced by: (None) |
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