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Mirrors > Home > MPE Home > Th. List > 4t4e16 | Structured version Visualization version GIF version |
Description: 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
4t4e16 | ⊢ (4 · 4) = ;16 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn0 12432 | . 2 ⊢ 4 ∈ ℕ0 | |
2 | 3nn0 12431 | . 2 ⊢ 3 ∈ ℕ0 | |
3 | df-4 12218 | . 2 ⊢ 4 = (3 + 1) | |
4 | 4t3e12 12716 | . 2 ⊢ (4 · 3) = ;12 | |
5 | 1nn0 12429 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 2nn0 12430 | . . 3 ⊢ 2 ∈ ℕ0 | |
7 | eqid 2736 | . . 3 ⊢ ;12 = ;12 | |
8 | 4cn 12238 | . . . 4 ⊢ 4 ∈ ℂ | |
9 | 2cn 12228 | . . . 4 ⊢ 2 ∈ ℂ | |
10 | 4p2e6 12306 | . . . 4 ⊢ (4 + 2) = 6 | |
11 | 8, 9, 10 | addcomli 11347 | . . 3 ⊢ (2 + 4) = 6 |
12 | 5, 6, 1, 7, 11 | decaddi 12678 | . 2 ⊢ (;12 + 4) = ;16 |
13 | 1, 2, 3, 4, 12 | 4t3lem 12715 | 1 ⊢ (4 · 4) = ;16 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 (class class class)co 7357 1c1 11052 · cmul 11056 2c2 12208 3c3 12209 4c4 12210 6c6 12212 ;cdc 12618 |
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 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 |
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 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-ltxr 11194 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-dec 12619 |
This theorem is referenced by: 2exp4 16957 2503lem2 17010 4001lem1 17013 4001lem2 17014 quart1lem 26205 quart1 26206 hgt750lem2 33265 3lexlogpow5ineq1 40511 aks4d1p1p7 40531 resqrtvalex 41907 wallispi2lem1 44302 fmtno4prmfac 45754 fmtno5faclem1 45761 2exp340mod341 45915 |
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