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 5059 mpbi 233 breqtri 5055 ltaddsubi 11279 remulcli 10735 2re 11790 3re 11796 9re 11815 eqcomi 2747 mvlladdi 10982 3cn 6cn 11807 eqtr3i 2763 6p3e9 11876 addcomi 10909 ltdiv1ii 11647 6re 11806 nngt0i 11755 2nn 11789 divcan3i 11464 recni 10733 2cn 11791 2ne0 11820 mpbir 234 eqtri 2761 mulcomi 10727 3t2e6 11882 divmuli 11472. (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 11790 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
3 | problem5.1 | . . . . . . 7 ⊢ 𝐴 ∈ ℝ | |
4 | 2, 3 | remulcli 10735 | . . . . . 6 ⊢ (2 · 𝐴) ∈ ℝ |
5 | 3re 11796 | . . . . . 6 ⊢ 3 ∈ ℝ | |
6 | 9re 11815 | . . . . . 6 ⊢ 9 ∈ ℝ | |
7 | 4, 5, 6 | ltaddsubi 11279 | . . . . 5 ⊢ (((2 · 𝐴) + 3) < 9 ↔ (2 · 𝐴) < (9 − 3)) |
8 | 1, 7 | mpbi 233 | . . . 4 ⊢ (2 · 𝐴) < (9 − 3) |
9 | 3cn 11797 | . . . . . 6 ⊢ 3 ∈ ℂ | |
10 | 6cn 11807 | . . . . . 6 ⊢ 6 ∈ ℂ | |
11 | 6p3e9 11876 | . . . . . . . 8 ⊢ (6 + 3) = 9 | |
12 | 10, 9 | addcomi 10909 | . . . . . . . 8 ⊢ (6 + 3) = (3 + 6) |
13 | 11, 12 | eqtr3i 2763 | . . . . . . 7 ⊢ 9 = (3 + 6) |
14 | 13 | eqcomi 2747 | . . . . . 6 ⊢ (3 + 6) = 9 |
15 | 9, 10, 14 | mvlladdi 10982 | . . . . 5 ⊢ 6 = (9 − 3) |
16 | 15 | eqcomi 2747 | . . . 4 ⊢ (9 − 3) = 6 |
17 | 8, 16 | breqtri 5055 | . . 3 ⊢ (2 · 𝐴) < 6 |
18 | 6re 11806 | . . . 4 ⊢ 6 ∈ ℝ | |
19 | 2nn 11789 | . . . . 5 ⊢ 2 ∈ ℕ | |
20 | 19 | nngt0i 11755 | . . . 4 ⊢ 0 < 2 |
21 | 4, 18, 2, 20 | ltdiv1ii 11647 | . . 3 ⊢ ((2 · 𝐴) < 6 ↔ ((2 · 𝐴) / 2) < (6 / 2)) |
22 | 17, 21 | mpbi 233 | . 2 ⊢ ((2 · 𝐴) / 2) < (6 / 2) |
23 | 3 | recni 10733 | . . 3 ⊢ 𝐴 ∈ ℂ |
24 | 2cn 11791 | . . 3 ⊢ 2 ∈ ℂ | |
25 | 2ne0 11820 | . . 3 ⊢ 2 ≠ 0 | |
26 | 23, 24, 25 | divcan3i 11464 | . 2 ⊢ ((2 · 𝐴) / 2) = 𝐴 |
27 | 24, 9 | mulcomi 10727 | . . . 4 ⊢ (2 · 3) = (3 · 2) |
28 | 3t2e6 11882 | . . . 4 ⊢ (3 · 2) = 6 | |
29 | 27, 28 | eqtri 2761 | . . 3 ⊢ (2 · 3) = 6 |
30 | 10, 24, 9, 25 | divmuli 11472 | . . 3 ⊢ ((6 / 2) = 3 ↔ (2 · 3) = 6) |
31 | 29, 30 | mpbir 234 | . 2 ⊢ (6 / 2) = 3 |
32 | 22, 26, 31 | 3brtr3i 5059 | 1 ⊢ 𝐴 < 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 class class class wbr 5030 (class class class)co 7170 ℝcr 10614 + caddc 10618 · cmul 10620 < clt 10753 − cmin 10948 / cdiv 11375 2c2 11771 3c3 11772 6c6 11775 9c9 11778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 |
This theorem is referenced by: (None) |
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