Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 8t8e64 | Structured version Visualization version GIF version |
Description: 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
8t8e64 | ⊢ (8 · 8) = ;64 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8nn0 11919 | . 2 ⊢ 8 ∈ ℕ0 | |
2 | 7nn0 11918 | . 2 ⊢ 7 ∈ ℕ0 | |
3 | df-8 11705 | . 2 ⊢ 8 = (7 + 1) | |
4 | 8t7e56 12217 | . 2 ⊢ (8 · 7) = ;56 | |
5 | 5nn0 11916 | . . 3 ⊢ 5 ∈ ℕ0 | |
6 | 6nn0 11917 | . . 3 ⊢ 6 ∈ ℕ0 | |
7 | eqid 2821 | . . 3 ⊢ ;56 = ;56 | |
8 | 5p1e6 11783 | . . 3 ⊢ (5 + 1) = 6 | |
9 | 4nn0 11915 | . . 3 ⊢ 4 ∈ ℕ0 | |
10 | 1 | nn0cni 11908 | . . . 4 ⊢ 8 ∈ ℂ |
11 | 6 | nn0cni 11908 | . . . 4 ⊢ 6 ∈ ℂ |
12 | 8p6e14 12181 | . . . 4 ⊢ (8 + 6) = ;14 | |
13 | 10, 11, 12 | addcomli 10831 | . . 3 ⊢ (6 + 8) = ;14 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 12158 | . 2 ⊢ (;56 + 8) = ;64 |
15 | 1, 2, 3, 4, 14 | 4t3lem 12194 | 1 ⊢ (8 · 8) = ;64 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7155 1c1 10537 · cmul 10541 4c4 11693 5c5 11694 6c6 11695 7c7 11696 8c8 11697 ;cdc 12097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-ltxr 10679 df-sub 10871 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-dec 12098 |
This theorem is referenced by: 2exp6 16421 1259lem4 16466 bposlem8 25866 bposlem9 25867 hgt750lem2 31923 8exp8mod9 43900 |
Copyright terms: Public domain | W3C validator |