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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 11393 | . . . . 5 ⊢ 3 ∈ ℝ | |
2 | 2re 11387 | . . . . 5 ⊢ 2 ∈ ℝ | |
3 | 2ne0 11424 | . . . . 5 ⊢ 2 ≠ 0 | |
4 | 1, 2, 3 | redivcli 11084 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
5 | 4 | recni 10343 | . . 3 ⊢ (3 / 2) ∈ ℂ |
6 | 1nn0 11598 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
7 | 5re 11402 | . . . . 5 ⊢ 5 ∈ ℝ | |
8 | dpcl 30115 | . . . . 5 ⊢ ((1 ∈ ℕ0 ∧ 5 ∈ ℝ) → (1.5) ∈ ℝ) | |
9 | 6, 7, 8 | mp2an 684 | . . . 4 ⊢ (1.5) ∈ ℝ |
10 | 9 | recni 10343 | . . 3 ⊢ (1.5) ∈ ℂ |
11 | 2cnne0 11530 | . . 3 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
12 | 5, 10, 11 | 3pm3.2i 1439 | . 2 ⊢ ((3 / 2) ∈ ℂ ∧ (1.5) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) |
13 | 5nn0 11602 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
14 | 3nn0 11600 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
15 | 0nn0 11597 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | eqid 2799 | . . . . . 6 ⊢ ;15 = ;15 | |
17 | df-2 11376 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
18 | 17 | oveq1i 6888 | . . . . . . 7 ⊢ (2 + 1) = ((1 + 1) + 1) |
19 | 2p1e3 11462 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
20 | 18, 19 | eqtr3i 2823 | . . . . . 6 ⊢ ((1 + 1) + 1) = 3 |
21 | 5p5e10 11856 | . . . . . 6 ⊢ (5 + 5) = ;10 | |
22 | 6, 13, 6, 13, 16, 16, 20, 15, 21 | decaddc 11839 | . . . . 5 ⊢ (;15 + ;15) = ;30 |
23 | 6, 13, 6, 13, 14, 15, 22 | dpadd 30135 | . . . 4 ⊢ ((1.5) + (1.5)) = (3.0) |
24 | 14 | dp0u 30125 | . . . 4 ⊢ (3.0) = 3 |
25 | 23, 24 | eqtri 2821 | . . 3 ⊢ ((1.5) + (1.5)) = 3 |
26 | 10 | times2i 11459 | . . 3 ⊢ ((1.5) · 2) = ((1.5) + (1.5)) |
27 | 1 | recni 10343 | . . . 4 ⊢ 3 ∈ ℂ |
28 | 11 | simpli 477 | . . . 4 ⊢ 2 ∈ ℂ |
29 | 27, 28, 3 | divcan1i 11061 | . . 3 ⊢ ((3 / 2) · 2) = 3 |
30 | 25, 26, 29 | 3eqtr4ri 2832 | . 2 ⊢ ((3 / 2) · 2) = ((1.5) · 2) |
31 | mulcan2 10957 | . . 3 ⊢ (((3 / 2) ∈ ℂ ∧ (1.5) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((3 / 2) · 2) = ((1.5) · 2) ↔ (3 / 2) = (1.5))) | |
32 | 31 | biimpa 469 | . 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 684 | 1 ⊢ (3 / 2) = (1.5) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 (class class class)co 6878 ℂcc 10222 ℝcr 10223 0cc0 10224 1c1 10225 + caddc 10227 · cmul 10229 / cdiv 10976 2c2 11368 3c3 11369 5c5 11371 ℕ0cn0 11580 ;cdc 11783 .cdp 30112 |
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 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
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 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-dec 11784 df-dp2 30096 df-dp 30113 |
This theorem is referenced by: (None) |
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