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Mirrors > Home > MPE Home > Th. List > 5t3e15 | Structured version Visualization version GIF version |
Description: 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
5t3e15 | ⊢ (5 · 3) = ;15 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 12488 | . 2 ⊢ 5 ∈ ℕ0 | |
2 | 2nn0 12485 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 12272 | . 2 ⊢ 3 = (2 + 1) | |
4 | 5t2e10 12773 | . 2 ⊢ (5 · 2) = ;10 | |
5 | dec10p 12716 | . 2 ⊢ (;10 + 5) = ;15 | |
6 | 1, 2, 3, 4, 5 | 4t3lem 12770 | 1 ⊢ (5 · 3) = ;15 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7404 0cc0 11106 1c1 11107 · cmul 11111 2c2 12263 3c3 12264 5c5 12266 ;cdc 12673 |
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 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-dec 12674 |
This theorem is referenced by: 5t4e20 12775 17prm 17046 prmlem2 17049 163prm 17054 317prm 17055 1259lem4 17063 2503lem2 17067 4001prm 17074 log2ub 26434 hgt750lem2 33602 3lexlogpow5ineq5 40863 inductionexd 42839 fmtno5lem2 46157 fmtno5lem3 46158 fmtno4prmfac193 46176 127prm 46202 |
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