Mathbox for Filip Cernatescu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > problem5 | Structured version Visualization version GIF version |
Description: Practice problem 5. Clues: 3brtr3i 5106 mpbi 229 breqtri 5102 ltaddsubi 11564 remulcli 11019 2re 12075 3re 12081 9re 12100 eqcomi 2742 mvlladdi 11267 3cn 6cn 12092 eqtr3i 2763 6p3e9 12161 addcomi 11194 ltdiv1ii 11932 6re 12091 nngt0i 12040 2nn 12074 divcan3i 11749 recni 11017 2cn 12076 2ne0 12105 mpbir 230 eqtri 2761 mulcomi 11011 3t2e6 12167 divmuli 11757. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
problem5.1 | ⊢ 𝐴 ∈ ℝ |
problem5.2 | ⊢ ((2 · 𝐴) + 3) < 9 |
Ref | Expression |
---|---|
problem5 | ⊢ 𝐴 < 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | problem5.2 | . . . . 5 ⊢ ((2 · 𝐴) + 3) < 9 | |
2 | 2re 12075 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
3 | problem5.1 | . . . . . . 7 ⊢ 𝐴 ∈ ℝ | |
4 | 2, 3 | remulcli 11019 | . . . . . 6 ⊢ (2 · 𝐴) ∈ ℝ |
5 | 3re 12081 | . . . . . 6 ⊢ 3 ∈ ℝ | |
6 | 9re 12100 | . . . . . 6 ⊢ 9 ∈ ℝ | |
7 | 4, 5, 6 | ltaddsubi 11564 | . . . . 5 ⊢ (((2 · 𝐴) + 3) < 9 ↔ (2 · 𝐴) < (9 − 3)) |
8 | 1, 7 | mpbi 229 | . . . 4 ⊢ (2 · 𝐴) < (9 − 3) |
9 | 3cn 12082 | . . . . . 6 ⊢ 3 ∈ ℂ | |
10 | 6cn 12092 | . . . . . 6 ⊢ 6 ∈ ℂ | |
11 | 6p3e9 12161 | . . . . . . . 8 ⊢ (6 + 3) = 9 | |
12 | 10, 9 | addcomi 11194 | . . . . . . . 8 ⊢ (6 + 3) = (3 + 6) |
13 | 11, 12 | eqtr3i 2763 | . . . . . . 7 ⊢ 9 = (3 + 6) |
14 | 13 | eqcomi 2742 | . . . . . 6 ⊢ (3 + 6) = 9 |
15 | 9, 10, 14 | mvlladdi 11267 | . . . . 5 ⊢ 6 = (9 − 3) |
16 | 15 | eqcomi 2742 | . . . 4 ⊢ (9 − 3) = 6 |
17 | 8, 16 | breqtri 5102 | . . 3 ⊢ (2 · 𝐴) < 6 |
18 | 6re 12091 | . . . 4 ⊢ 6 ∈ ℝ | |
19 | 2nn 12074 | . . . . 5 ⊢ 2 ∈ ℕ | |
20 | 19 | nngt0i 12040 | . . . 4 ⊢ 0 < 2 |
21 | 4, 18, 2, 20 | ltdiv1ii 11932 | . . 3 ⊢ ((2 · 𝐴) < 6 ↔ ((2 · 𝐴) / 2) < (6 / 2)) |
22 | 17, 21 | mpbi 229 | . 2 ⊢ ((2 · 𝐴) / 2) < (6 / 2) |
23 | 3 | recni 11017 | . . 3 ⊢ 𝐴 ∈ ℂ |
24 | 2cn 12076 | . . 3 ⊢ 2 ∈ ℂ | |
25 | 2ne0 12105 | . . 3 ⊢ 2 ≠ 0 | |
26 | 23, 24, 25 | divcan3i 11749 | . 2 ⊢ ((2 · 𝐴) / 2) = 𝐴 |
27 | 24, 9 | mulcomi 11011 | . . . 4 ⊢ (2 · 3) = (3 · 2) |
28 | 3t2e6 12167 | . . . 4 ⊢ (3 · 2) = 6 | |
29 | 27, 28 | eqtri 2761 | . . 3 ⊢ (2 · 3) = 6 |
30 | 10, 24, 9, 25 | divmuli 11757 | . . 3 ⊢ ((6 / 2) = 3 ↔ (2 · 3) = 6) |
31 | 29, 30 | mpbir 230 | . 2 ⊢ (6 / 2) = 3 |
32 | 22, 26, 31 | 3brtr3i 5106 | 1 ⊢ 𝐴 < 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2101 class class class wbr 5077 (class class class)co 7295 ℝcr 10898 + caddc 10902 · cmul 10904 < clt 11037 − cmin 11233 / cdiv 11660 2c2 12056 3c3 12057 6c6 12060 9c9 12063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 |
This theorem is referenced by: (None) |
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