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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 11726 | . 2 ⊢ 4 ∈ ℕ0 | |
2 | 2nn0 11724 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 11502 | . 2 ⊢ 3 = (2 + 1) | |
4 | 4t2e8 11613 | . 2 ⊢ (4 · 2) = 8 | |
5 | 8p4e12 11993 | . 2 ⊢ (8 + 4) = ;12 | |
6 | 1, 2, 3, 4, 5 | 4t3lem 12008 | 1 ⊢ (4 · 3) = ;12 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 (class class class)co 6974 1c1 10334 · cmul 10338 2c2 11493 3c3 11494 4c4 11495 8c8 11499 ;cdc 11909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-ltxr 10477 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-dec 11910 |
This theorem is referenced by: 4t4e16 12010 13prm 16303 43prm 16309 139prm 16311 163prm 16312 317prm 16313 631prm 16314 1259lem4 16321 1259prm 16323 2503lem1 16324 2503lem2 16325 4001lem2 16329 4001lem4 16331 quartlem1 25151 hgt750lem2 31603 fmtno4prmfac 43136 fmtno4prmfac193 43137 2exp340mod341 43300 |
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