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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 12325 | . . . . 5 ⊢ 3 ∈ ℝ | |
2 | 2re 12319 | . . . . 5 ⊢ 2 ∈ ℝ | |
3 | 2ne0 12349 | . . . . 5 ⊢ 2 ≠ 0 | |
4 | 1, 2, 3 | redivcli 12014 | . . . 4 ⊢ (3 / 2) ∈ ℝ |
5 | 4 | recni 11260 | . . 3 ⊢ (3 / 2) ∈ ℂ |
6 | 1nn0 12521 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
7 | 5re 12332 | . . . . 5 ⊢ 5 ∈ ℝ | |
8 | dpcl 32699 | . . . . 5 ⊢ ((1 ∈ ℕ0 ∧ 5 ∈ ℝ) → (1.5) ∈ ℝ) | |
9 | 6, 7, 8 | mp2an 690 | . . . 4 ⊢ (1.5) ∈ ℝ |
10 | 9 | recni 11260 | . . 3 ⊢ (1.5) ∈ ℂ |
11 | 2cnne0 12455 | . . 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 12525 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
14 | 3nn0 12523 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
15 | 0nn0 12520 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | eqid 2725 | . . . . . 6 ⊢ ;15 = ;15 | |
17 | df-2 12308 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
18 | 17 | oveq1i 7429 | . . . . . . 7 ⊢ (2 + 1) = ((1 + 1) + 1) |
19 | 2p1e3 12387 | . . . . . . 7 ⊢ (2 + 1) = 3 | |
20 | 18, 19 | eqtr3i 2755 | . . . . . 6 ⊢ ((1 + 1) + 1) = 3 |
21 | 5p5e10 12781 | . . . . . 6 ⊢ (5 + 5) = ;10 | |
22 | 6, 13, 6, 13, 16, 16, 20, 15, 21 | decaddc 12765 | . . . . 5 ⊢ (;15 + ;15) = ;30 |
23 | 6, 13, 6, 13, 14, 15, 22 | dpadd 32719 | . . . 4 ⊢ ((1.5) + (1.5)) = (3.0) |
24 | 14 | dp0u 32709 | . . . 4 ⊢ (3.0) = 3 |
25 | 23, 24 | eqtri 2753 | . . 3 ⊢ ((1.5) + (1.5)) = 3 |
26 | 10 | times2i 12384 | . . 3 ⊢ ((1.5) · 2) = ((1.5) + (1.5)) |
27 | 1 | recni 11260 | . . . 4 ⊢ 3 ∈ ℂ |
28 | 11 | simpli 482 | . . . 4 ⊢ 2 ∈ ℂ |
29 | 27, 28, 3 | divcan1i 11991 | . . 3 ⊢ ((3 / 2) · 2) = 3 |
30 | 25, 26, 29 | 3eqtr4ri 2764 | . 2 ⊢ ((3 / 2) · 2) = ((1.5) · 2) |
31 | mulcan2 11884 | . . 3 ⊢ (((3 / 2) ∈ ℂ ∧ (1.5) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((3 / 2) · 2) = ((1.5) · 2) ↔ (3 / 2) = (1.5))) | |
32 | 31 | biimpa 475 | . 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 690 | 1 ⊢ (3 / 2) = (1.5) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 / cdiv 11903 2c2 12300 3c3 12301 5c5 12303 ℕ0cn0 12505 ;cdc 12710 .cdp 32696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-dec 12711 df-dp2 32680 df-dp 32697 |
This theorem is referenced by: (None) |
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