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| Mirrors > Home > MPE Home > Th. List > 6t3e18 | Structured version Visualization version GIF version | ||
| Description: 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) |
| Ref | Expression |
|---|---|
| 6t3e18 | ⊢ (6 · 3) = ;18 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 6nn0 12547 | . 2 ⊢ 6 ∈ ℕ0 | |
| 2 | 2nn0 12543 | . 2 ⊢ 2 ∈ ℕ0 | |
| 3 | df-3 12330 | . 2 ⊢ 3 = (2 + 1) | |
| 4 | 6t2e12 12837 | . 2 ⊢ (6 · 2) = ;12 | |
| 5 | 1nn0 12542 | . . 3 ⊢ 1 ∈ ℕ0 | |
| 6 | eqid 2737 | . . 3 ⊢ ;12 = ;12 | |
| 7 | 6cn 12357 | . . . 4 ⊢ 6 ∈ ℂ | |
| 8 | 2cn 12341 | . . . 4 ⊢ 2 ∈ ℂ | |
| 9 | 6p2e8 12425 | . . . 4 ⊢ (6 + 2) = 8 | |
| 10 | 7, 8, 9 | addcomli 11453 | . . 3 ⊢ (2 + 6) = 8 |
| 11 | 5, 2, 1, 6, 10 | decaddi 12793 | . 2 ⊢ (;12 + 6) = ;18 |
| 12 | 1, 2, 3, 4, 11 | 4t3lem 12830 | 1 ⊢ (6 · 3) = ;18 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7431 1c1 11156 · cmul 11160 2c2 12321 3c3 12322 6c6 12325 8c8 12327 ;cdc 12733 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 |
| This theorem is referenced by: 6t4e24 12839 19prm 17155 83prm 17160 139prm 17161 1259lem2 17169 1259lem4 17171 ex-lcm 30477 fmtno5lem1 47540 fmtno5lem3 47542 fmtno4prmfac 47559 139prmALT 47583 |
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