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Mirrors > Home > MPE Home > Th. List > 9t3e27 | Structured version Visualization version GIF version |
Description: 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9t3e27 | ⊢ (9 · 3) = ;27 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 12436 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 2nn0 12429 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 12216 | . 2 ⊢ 3 = (2 + 1) | |
4 | 9t2e18 12739 | . 2 ⊢ (9 · 2) = ;18 | |
5 | 1nn0 12428 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 8nn0 12435 | . . 3 ⊢ 8 ∈ ℕ0 | |
7 | eqid 2736 | . . 3 ⊢ ;18 = ;18 | |
8 | 1p1e2 12277 | . . 3 ⊢ (1 + 1) = 2 | |
9 | 7nn0 12434 | . . 3 ⊢ 7 ∈ ℕ0 | |
10 | 1 | nn0cni 12424 | . . . 4 ⊢ 9 ∈ ℂ |
11 | 6 | nn0cni 12424 | . . . 4 ⊢ 8 ∈ ℂ |
12 | 9p8e17 12710 | . . . 4 ⊢ (9 + 8) = ;17 | |
13 | 10, 11, 12 | addcomli 11346 | . . 3 ⊢ (8 + 9) = ;17 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 12678 | . 2 ⊢ (;18 + 9) = ;27 |
15 | 1, 2, 3, 4, 14 | 4t3lem 12714 | 1 ⊢ (9 · 3) = ;27 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 (class class class)co 7356 1c1 11051 · cmul 11055 2c2 12207 3c3 12208 7c7 12212 8c8 12213 9c9 12214 ;cdc 12617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-ltxr 11193 df-sub 11386 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-dec 12618 |
This theorem is referenced by: 9t4e36 12741 3exp3 16963 prmlem2 16991 631prm 16998 1259lem4 17005 2503lem2 17009 4001lem3 17014 mcubic 26195 log2ublem3 26296 log2ub 26297 3exp7 40500 3lexlogpow2ineq2 40506 257prm 45724 fmtno4nprmfac193 45737 127prm 45762 m11nprm 45764 |
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