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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2t6m3t4e0 | Structured version Visualization version GIF version |
Description: 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.) |
Ref | Expression |
---|---|
2t6m3t4e0 | ⊢ ((2 · 6) − (3 · 4)) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6cn 12252 | . . . 4 ⊢ 6 ∈ ℂ | |
2 | 1 | 2timesi 12299 | . . 3 ⊢ (2 · 6) = (6 + 6) |
3 | 2p2e4 12296 | . . . . . 6 ⊢ (2 + 2) = 4 | |
4 | 3 | eqcomi 2742 | . . . . 5 ⊢ 4 = (2 + 2) |
5 | 4 | oveq2i 7372 | . . . 4 ⊢ (3 · 4) = (3 · (2 + 2)) |
6 | 3cn 12242 | . . . . 5 ⊢ 3 ∈ ℂ | |
7 | 2cn 12236 | . . . . 5 ⊢ 2 ∈ ℂ | |
8 | 6, 7, 7 | adddii 11175 | . . . 4 ⊢ (3 · (2 + 2)) = ((3 · 2) + (3 · 2)) |
9 | 3t2e6 12327 | . . . . 5 ⊢ (3 · 2) = 6 | |
10 | 9, 9 | oveq12i 7373 | . . . 4 ⊢ ((3 · 2) + (3 · 2)) = (6 + 6) |
11 | 5, 8, 10 | 3eqtri 2765 | . . 3 ⊢ (3 · 4) = (6 + 6) |
12 | 2, 11 | oveq12i 7373 | . 2 ⊢ ((2 · 6) − (3 · 4)) = ((6 + 6) − (6 + 6)) |
13 | 1, 1 | addcli 11169 | . . 3 ⊢ (6 + 6) ∈ ℂ |
14 | 13 | subidi 11480 | . 2 ⊢ ((6 + 6) − (6 + 6)) = 0 |
15 | 12, 14 | eqtri 2761 | 1 ⊢ ((2 · 6) − (3 · 4)) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7361 0cc0 11059 + caddc 11062 · cmul 11064 − cmin 11393 2c2 12216 3c3 12217 4c4 12218 6c6 12220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-ltxr 11202 df-sub 11395 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 |
This theorem is referenced by: zlmodzxzequa 46667 zlmodzxzequap 46670 |
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