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Mirrors > Home > MPE Home > Th. List > 4t3e12 | Structured version Visualization version GIF version |
Description: 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
4t3e12 | ⊢ (4 · 3) = ;12 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn0 12429 | . 2 ⊢ 4 ∈ ℕ0 | |
2 | 2nn0 12427 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 12214 | . 2 ⊢ 3 = (2 + 1) | |
4 | 4t2e8 12318 | . 2 ⊢ (4 · 2) = 8 | |
5 | 8p4e12 12697 | . 2 ⊢ (8 + 4) = ;12 | |
6 | 1, 2, 3, 4, 5 | 4t3lem 12712 | 1 ⊢ (4 · 3) = ;12 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 (class class class)co 7354 1c1 11049 · cmul 11053 2c2 12205 3c3 12206 4c4 12207 8c8 12211 ;cdc 12615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7357 df-om 7800 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-ltxr 11191 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-dec 12616 |
This theorem is referenced by: 4t4e16 12714 13prm 16985 43prm 16991 139prm 16993 163prm 16994 317prm 16995 631prm 16996 1259lem4 17003 1259prm 17005 2503lem1 17006 2503lem2 17007 4001lem2 17011 4001lem4 17013 quartlem1 26203 hgt750lem2 33156 fmtno4prmfac 45734 fmtno4prmfac193 45735 2exp340mod341 45895 |
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