| Mathbox for Filip Cernatescu |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > problem2 | Structured version Visualization version GIF version | ||
| Description: Practice problem 2. Clues: oveq12i 7425 adddiri 11256 add4i 11468 mulcli 11250 recni 11257 2re 12322 3eqtri 2761 10re 12735 5re 12335 1re 11243 4re 12332 eqcomi 2743 5p4e9 12406 oveq1i 7423 df-3 12312. (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 12322 | . . . . 5 ⊢ 2 ∈ ℝ | |
| 2 | 1 | recni 11257 | . . . 4 ⊢ 2 ∈ ℂ |
| 3 | 10re 12735 | . . . . 5 ⊢ ;10 ∈ ℝ | |
| 4 | 3 | recni 11257 | . . . 4 ⊢ ;10 ∈ ℂ |
| 5 | 2, 4 | mulcli 11250 | . . 3 ⊢ (2 · ;10) ∈ ℂ |
| 6 | 5re 12335 | . . . 4 ⊢ 5 ∈ ℝ | |
| 7 | 6 | recni 11257 | . . 3 ⊢ 5 ∈ ℂ |
| 8 | 1re 11243 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 9 | 8 | recni 11257 | . . . 4 ⊢ 1 ∈ ℂ |
| 10 | 9, 4 | mulcli 11250 | . . 3 ⊢ (1 · ;10) ∈ ℂ |
| 11 | 4re 12332 | . . . 4 ⊢ 4 ∈ ℝ | |
| 12 | 11 | recni 11257 | . . 3 ⊢ 4 ∈ ℂ |
| 13 | 5, 7, 10, 12 | add4i 11468 | . 2 ⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = (((2 · ;10) + (1 · ;10)) + (5 + 4)) |
| 14 | 2, 9, 4 | adddiri 11256 | . . . 4 ⊢ ((2 + 1) · ;10) = ((2 · ;10) + (1 · ;10)) |
| 15 | 14 | eqcomi 2743 | . . 3 ⊢ ((2 · ;10) + (1 · ;10)) = ((2 + 1) · ;10) |
| 16 | 5p4e9 12406 | . . 3 ⊢ (5 + 4) = 9 | |
| 17 | 15, 16 | oveq12i 7425 | . 2 ⊢ (((2 · ;10) + (1 · ;10)) + (5 + 4)) = (((2 + 1) · ;10) + 9) |
| 18 | df-3 12312 | . . . . 5 ⊢ 3 = (2 + 1) | |
| 19 | 18 | eqcomi 2743 | . . . 4 ⊢ (2 + 1) = 3 |
| 20 | 19 | oveq1i 7423 | . . 3 ⊢ ((2 + 1) · ;10) = (3 · ;10) |
| 21 | 20 | oveq1i 7423 | . 2 ⊢ (((2 + 1) · ;10) + 9) = ((3 · ;10) + 9) |
| 22 | 13, 17, 21 | 3eqtri 2761 | 1 ⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1539 (class class class)co 7413 0cc0 11137 1c1 11138 + caddc 11140 · cmul 11142 2c2 12303 3c3 12304 4c4 12305 5c5 12306 9c9 12310 ;cdc 12716 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-po 5572 df-so 5573 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7416 df-er 8727 df-en 8968 df-dom 8969 df-sdom 8970 df-pnf 11279 df-mnf 11280 df-ltxr 11282 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-dec 12717 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |