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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 7147 adddiri 10643 add4i 10853 mulcli 10637 recni 10644 2re 11699 3eqtri 2825 10re 12105 5re 11712 1re 10630 4re 11709 eqcomi 2807 5p4e9 11783 oveq1i 7145 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 10644 | . . . 4 ⊢ 2 ∈ ℂ |
3 | 10re 12105 | . . . . 5 ⊢ ;10 ∈ ℝ | |
4 | 3 | recni 10644 | . . . 4 ⊢ ;10 ∈ ℂ |
5 | 2, 4 | mulcli 10637 | . . 3 ⊢ (2 · ;10) ∈ ℂ |
6 | 5re 11712 | . . . 4 ⊢ 5 ∈ ℝ | |
7 | 6 | recni 10644 | . . 3 ⊢ 5 ∈ ℂ |
8 | 1re 10630 | . . . . 5 ⊢ 1 ∈ ℝ | |
9 | 8 | recni 10644 | . . . 4 ⊢ 1 ∈ ℂ |
10 | 9, 4 | mulcli 10637 | . . 3 ⊢ (1 · ;10) ∈ ℂ |
11 | 4re 11709 | . . . 4 ⊢ 4 ∈ ℝ | |
12 | 11 | recni 10644 | . . 3 ⊢ 4 ∈ ℂ |
13 | 5, 7, 10, 12 | add4i 10853 | . 2 ⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = (((2 · ;10) + (1 · ;10)) + (5 + 4)) |
14 | 2, 9, 4 | adddiri 10643 | . . . 4 ⊢ ((2 + 1) · ;10) = ((2 · ;10) + (1 · ;10)) |
15 | 14 | eqcomi 2807 | . . 3 ⊢ ((2 · ;10) + (1 · ;10)) = ((2 + 1) · ;10) |
16 | 5p4e9 11783 | . . 3 ⊢ (5 + 4) = 9 | |
17 | 15, 16 | oveq12i 7147 | . 2 ⊢ (((2 · ;10) + (1 · ;10)) + (5 + 4)) = (((2 + 1) · ;10) + 9) |
18 | df-3 11689 | . . . . 5 ⊢ 3 = (2 + 1) | |
19 | 18 | eqcomi 2807 | . . . 4 ⊢ (2 + 1) = 3 |
20 | 19 | oveq1i 7145 | . . 3 ⊢ ((2 + 1) · ;10) = (3 · ;10) |
21 | 20 | oveq1i 7145 | . 2 ⊢ (((2 + 1) · ;10) + 9) = ((3 · ;10) + 9) |
22 | 13, 17, 21 | 3eqtri 2825 | 1 ⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 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 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 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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