| 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 7401 adddiri 11193 add4i 11405 mulcli 11187 recni 11194 2re 12261 3eqtri 2757 10re 12674 5re 12274 1re 11180 4re 12271 eqcomi 2739 5p4e9 12345 oveq1i 7399 df-3 12251. (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 12261 | . . . . 5 ⊢ 2 ∈ ℝ | |
| 2 | 1 | recni 11194 | . . . 4 ⊢ 2 ∈ ℂ |
| 3 | 10re 12674 | . . . . 5 ⊢ ;10 ∈ ℝ | |
| 4 | 3 | recni 11194 | . . . 4 ⊢ ;10 ∈ ℂ |
| 5 | 2, 4 | mulcli 11187 | . . 3 ⊢ (2 · ;10) ∈ ℂ |
| 6 | 5re 12274 | . . . 4 ⊢ 5 ∈ ℝ | |
| 7 | 6 | recni 11194 | . . 3 ⊢ 5 ∈ ℂ |
| 8 | 1re 11180 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 9 | 8 | recni 11194 | . . . 4 ⊢ 1 ∈ ℂ |
| 10 | 9, 4 | mulcli 11187 | . . 3 ⊢ (1 · ;10) ∈ ℂ |
| 11 | 4re 12271 | . . . 4 ⊢ 4 ∈ ℝ | |
| 12 | 11 | recni 11194 | . . 3 ⊢ 4 ∈ ℂ |
| 13 | 5, 7, 10, 12 | add4i 11405 | . 2 ⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = (((2 · ;10) + (1 · ;10)) + (5 + 4)) |
| 14 | 2, 9, 4 | adddiri 11193 | . . . 4 ⊢ ((2 + 1) · ;10) = ((2 · ;10) + (1 · ;10)) |
| 15 | 14 | eqcomi 2739 | . . 3 ⊢ ((2 · ;10) + (1 · ;10)) = ((2 + 1) · ;10) |
| 16 | 5p4e9 12345 | . . 3 ⊢ (5 + 4) = 9 | |
| 17 | 15, 16 | oveq12i 7401 | . 2 ⊢ (((2 · ;10) + (1 · ;10)) + (5 + 4)) = (((2 + 1) · ;10) + 9) |
| 18 | df-3 12251 | . . . . 5 ⊢ 3 = (2 + 1) | |
| 19 | 18 | eqcomi 2739 | . . . 4 ⊢ (2 + 1) = 3 |
| 20 | 19 | oveq1i 7399 | . . 3 ⊢ ((2 + 1) · ;10) = (3 · ;10) |
| 21 | 20 | oveq1i 7399 | . 2 ⊢ (((2 + 1) · ;10) + 9) = ((3 · ;10) + 9) |
| 22 | 13, 17, 21 | 3eqtri 2757 | 1 ⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7389 0cc0 11074 1c1 11075 + caddc 11077 · cmul 11079 2c2 12242 3c3 12243 4c4 12244 5c5 12245 9c9 12249 ;cdc 12655 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-po 5548 df-so 5549 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-ov 7392 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11216 df-mnf 11217 df-ltxr 11219 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-dec 12656 |
| This theorem is referenced by: (None) |
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