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Mirrors > Home > MPE Home > Th. List > 9t8e72 | Structured version Visualization version GIF version |
Description: 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t8e72 | ⊢ (9 · 8) = ;72 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 11606 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 7nn0 11604 | . 2 ⊢ 7 ∈ ℕ0 | |
3 | df-8 11382 | . 2 ⊢ 8 = (7 + 1) | |
4 | 9t7e63 11912 | . 2 ⊢ (9 · 7) = ;63 | |
5 | 6nn0 11603 | . . 3 ⊢ 6 ∈ ℕ0 | |
6 | 3nn0 11600 | . . 3 ⊢ 3 ∈ ℕ0 | |
7 | eqid 2799 | . . 3 ⊢ ;63 = ;63 | |
8 | 6p1e7 11468 | . . 3 ⊢ (6 + 1) = 7 | |
9 | 2nn0 11599 | . . 3 ⊢ 2 ∈ ℕ0 | |
10 | 9cn 11419 | . . . 4 ⊢ 9 ∈ ℂ | |
11 | 3cn 11394 | . . . 4 ⊢ 3 ∈ ℂ | |
12 | 9p3e12 11873 | . . . 4 ⊢ (9 + 3) = ;12 | |
13 | 10, 11, 12 | addcomli 10518 | . . 3 ⊢ (3 + 9) = ;12 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 11845 | . 2 ⊢ (;63 + 9) = ;72 |
15 | 1, 2, 3, 4, 14 | 4t3lem 11882 | 1 ⊢ (9 · 8) = ;72 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 (class class class)co 6878 1c1 10225 · cmul 10229 2c2 11368 3c3 11369 6c6 11372 7c7 11373 8c8 11374 9c9 11375 ;cdc 11783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-sub 10558 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-dec 11784 |
This theorem is referenced by: 9t9e81 11914 163prm 16159 2503prm 16174 quartlem1 24936 log2ublem3 25027 |
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