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Mirrors > Home > MPE Home > Th. List > 9t5e45 | Structured version Visualization version GIF version |
Description: 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t5e45 | ⊢ (9 · 5) = ;45 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 11651 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 4nn0 11646 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | df-5 11424 | . 2 ⊢ 5 = (4 + 1) | |
4 | 9t4e36 11954 | . 2 ⊢ (9 · 4) = ;36 | |
5 | 3nn0 11645 | . . 3 ⊢ 3 ∈ ℕ0 | |
6 | 6nn0 11648 | . . 3 ⊢ 6 ∈ ℕ0 | |
7 | eqid 2825 | . . 3 ⊢ ;36 = ;36 | |
8 | 3p1e4 11510 | . . 3 ⊢ (3 + 1) = 4 | |
9 | 5nn0 11647 | . . 3 ⊢ 5 ∈ ℕ0 | |
10 | 1 | nn0cni 11638 | . . . 4 ⊢ 9 ∈ ℂ |
11 | 6 | nn0cni 11638 | . . . 4 ⊢ 6 ∈ ℂ |
12 | 9p6e15 11921 | . . . 4 ⊢ (9 + 6) = ;15 | |
13 | 10, 11, 12 | addcomli 10554 | . . 3 ⊢ (6 + 9) = ;15 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 11890 | . 2 ⊢ (;36 + 9) = ;45 |
15 | 1, 2, 3, 4, 14 | 4t3lem 11927 | 1 ⊢ (9 · 5) = ;45 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 (class class class)co 6910 1c1 10260 · cmul 10264 3c3 11414 4c4 11415 5c5 11416 6c6 11417 9c9 11420 ;cdc 11828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-ltxr 10403 df-sub 10594 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-dec 11829 |
This theorem is referenced by: 9t6e54 11956 1259lem1 16210 2503lem2 16217 4001lem1 16220 log2ublem3 25095 |
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