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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 11903 | . 2 ⊢ 4 ∈ ℕ0 | |
2 | 3nn0 11902 | . 2 ⊢ 3 ∈ ℕ0 | |
3 | df-4 11689 | . 2 ⊢ 4 = (3 + 1) | |
4 | 4t3e12 12183 | . 2 ⊢ (4 · 3) = ;12 | |
5 | 1nn0 11900 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 2nn0 11901 | . . 3 ⊢ 2 ∈ ℕ0 | |
7 | eqid 2821 | . . 3 ⊢ ;12 = ;12 | |
8 | 4cn 11709 | . . . 4 ⊢ 4 ∈ ℂ | |
9 | 2cn 11699 | . . . 4 ⊢ 2 ∈ ℂ | |
10 | 4p2e6 11777 | . . . 4 ⊢ (4 + 2) = 6 | |
11 | 8, 9, 10 | addcomli 10818 | . . 3 ⊢ (2 + 4) = 6 |
12 | 5, 6, 1, 7, 11 | decaddi 12145 | . 2 ⊢ (;12 + 4) = ;16 |
13 | 1, 2, 3, 4, 12 | 4t3lem 12182 | 1 ⊢ (4 · 4) = ;16 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7142 1c1 10524 · cmul 10528 2c2 11679 3c3 11680 4c4 11681 6c6 11683 ;cdc 12085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-om 7567 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-ltxr 10666 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-dec 12086 |
This theorem is referenced by: 2exp4 16404 2503lem2 16454 4001lem1 16457 4001lem2 16458 quart1lem 25419 quart1 25420 hgt750lem2 31930 wallispi2lem1 42446 fmtno4prmfac 43819 fmtno5faclem1 43826 2exp340mod341 43983 |
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