Mathbox for Filip Cernatescu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > problem2 | Structured version Visualization version GIF version |
Description: Practice problem 2. Clues: oveq12i 7157 adddiri 10642 add4i 10852 mulcli 10636 recni 10643 2re 11699 3eqtri 2845 10re 12105 5re 11712 1re 10629 4re 11709 eqcomi 2827 5p4e9 11783 oveq1i 7155 df-3 11689. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, 9-Sep-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
problem2 | ⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 11699 | . . . . 5 ⊢ 2 ∈ ℝ | |
2 | 1 | recni 10643 | . . . 4 ⊢ 2 ∈ ℂ |
3 | 10re 12105 | . . . . 5 ⊢ ;10 ∈ ℝ | |
4 | 3 | recni 10643 | . . . 4 ⊢ ;10 ∈ ℂ |
5 | 2, 4 | mulcli 10636 | . . 3 ⊢ (2 · ;10) ∈ ℂ |
6 | 5re 11712 | . . . 4 ⊢ 5 ∈ ℝ | |
7 | 6 | recni 10643 | . . 3 ⊢ 5 ∈ ℂ |
8 | 1re 10629 | . . . . 5 ⊢ 1 ∈ ℝ | |
9 | 8 | recni 10643 | . . . 4 ⊢ 1 ∈ ℂ |
10 | 9, 4 | mulcli 10636 | . . 3 ⊢ (1 · ;10) ∈ ℂ |
11 | 4re 11709 | . . . 4 ⊢ 4 ∈ ℝ | |
12 | 11 | recni 10643 | . . 3 ⊢ 4 ∈ ℂ |
13 | 5, 7, 10, 12 | add4i 10852 | . 2 ⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = (((2 · ;10) + (1 · ;10)) + (5 + 4)) |
14 | 2, 9, 4 | adddiri 10642 | . . . 4 ⊢ ((2 + 1) · ;10) = ((2 · ;10) + (1 · ;10)) |
15 | 14 | eqcomi 2827 | . . 3 ⊢ ((2 · ;10) + (1 · ;10)) = ((2 + 1) · ;10) |
16 | 5p4e9 11783 | . . 3 ⊢ (5 + 4) = 9 | |
17 | 15, 16 | oveq12i 7157 | . 2 ⊢ (((2 · ;10) + (1 · ;10)) + (5 + 4)) = (((2 + 1) · ;10) + 9) |
18 | df-3 11689 | . . . . 5 ⊢ 3 = (2 + 1) | |
19 | 18 | eqcomi 2827 | . . . 4 ⊢ (2 + 1) = 3 |
20 | 19 | oveq1i 7155 | . . 3 ⊢ ((2 + 1) · ;10) = (3 · ;10) |
21 | 20 | oveq1i 7155 | . 2 ⊢ (((2 + 1) · ;10) + 9) = ((3 · ;10) + 9) |
22 | 13, 17, 21 | 3eqtri 2845 | 1 ⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 2c2 11680 3c3 11681 4c4 11682 5c5 11683 9c9 11687 ;cdc 12086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-dec 12087 |
This theorem is referenced by: (None) |
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