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Mirrors > Home > MPE Home > Th. List > 9t9e81 | Structured version Visualization version GIF version |
Description: 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t9e81 | ⊢ (9 · 9) = ;81 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 11522 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 8nn0 11521 | . 2 ⊢ 8 ∈ ℕ0 | |
3 | df-9 11291 | . 2 ⊢ 9 = (8 + 1) | |
4 | 9t8e72 11874 | . 2 ⊢ (9 · 8) = ;72 | |
5 | 7nn0 11520 | . . 3 ⊢ 7 ∈ ℕ0 | |
6 | 2nn0 11515 | . . 3 ⊢ 2 ∈ ℕ0 | |
7 | eqid 2771 | . . 3 ⊢ ;72 = ;72 | |
8 | 7p1e8 11363 | . . 3 ⊢ (7 + 1) = 8 | |
9 | 1nn0 11514 | . . 3 ⊢ 1 ∈ ℕ0 | |
10 | 9cn 11313 | . . . 4 ⊢ 9 ∈ ℂ | |
11 | 2cn 11296 | . . . 4 ⊢ 2 ∈ ℂ | |
12 | 9p2e11 11824 | . . . 4 ⊢ (9 + 2) = ;11 | |
13 | 10, 11, 12 | addcomli 10433 | . . 3 ⊢ (2 + 9) = ;11 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 11785 | . 2 ⊢ (;72 + 9) = ;81 |
15 | 1, 2, 3, 4, 14 | 4t3lem 11836 | 1 ⊢ (9 · 9) = ;81 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 (class class class)co 6795 1c1 10142 · cmul 10146 2c2 11275 7c7 11280 8c8 11281 9c9 11282 ;cdc 11699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-om 7216 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-er 7899 df-en 8113 df-dom 8114 df-sdom 8115 df-pnf 10281 df-mnf 10282 df-ltxr 10284 df-sub 10473 df-nn 11226 df-2 11284 df-3 11285 df-4 11286 df-5 11287 df-6 11288 df-7 11289 df-8 11290 df-9 11291 df-n0 11499 df-dec 11700 |
This theorem is referenced by: prmlem2 16033 2503lem2 16051 4001lem1 16054 4001lem2 16055 log2ublem3 24895 hgt750lem2 31069 fmtno4nprmfac193 42011 3exp4mod41 42058 |
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