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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 12544 | . 2 ⊢ 6 ∈ ℕ0 | |
2 | 2nn0 12540 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 12327 | . 2 ⊢ 3 = (2 + 1) | |
4 | 6t2e12 12834 | . 2 ⊢ (6 · 2) = ;12 | |
5 | 1nn0 12539 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | eqid 2734 | . . 3 ⊢ ;12 = ;12 | |
7 | 6cn 12354 | . . . 4 ⊢ 6 ∈ ℂ | |
8 | 2cn 12338 | . . . 4 ⊢ 2 ∈ ℂ | |
9 | 6p2e8 12422 | . . . 4 ⊢ (6 + 2) = 8 | |
10 | 7, 8, 9 | addcomli 11450 | . . 3 ⊢ (2 + 6) = 8 |
11 | 5, 2, 1, 6, 10 | decaddi 12790 | . 2 ⊢ (;12 + 6) = ;18 |
12 | 1, 2, 3, 4, 11 | 4t3lem 12827 | 1 ⊢ (6 · 3) = ;18 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 (class class class)co 7430 1c1 11153 · cmul 11157 2c2 12318 3c3 12319 6c6 12322 8c8 12324 ;cdc 12730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-dec 12731 |
This theorem is referenced by: 6t4e24 12836 19prm 17151 83prm 17156 139prm 17157 1259lem2 17165 1259lem4 17167 ex-lcm 30486 fmtno5lem1 47477 fmtno5lem3 47479 fmtno4prmfac 47496 139prmALT 47520 |
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