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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 11775 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 2nn0 11768 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 11555 | . 2 ⊢ 3 = (2 + 1) | |
4 | 9t2e18 12074 | . 2 ⊢ (9 · 2) = ;18 | |
5 | 1nn0 11767 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 8nn0 11774 | . . 3 ⊢ 8 ∈ ℕ0 | |
7 | eqid 2797 | . . 3 ⊢ ;18 = ;18 | |
8 | 1p1e2 11616 | . . 3 ⊢ (1 + 1) = 2 | |
9 | 7nn0 11773 | . . 3 ⊢ 7 ∈ ℕ0 | |
10 | 1 | nn0cni 11763 | . . . 4 ⊢ 9 ∈ ℂ |
11 | 6 | nn0cni 11763 | . . . 4 ⊢ 8 ∈ ℂ |
12 | 9p8e17 12045 | . . . 4 ⊢ (9 + 8) = ;17 | |
13 | 10, 11, 12 | addcomli 10685 | . . 3 ⊢ (8 + 9) = ;17 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 12013 | . 2 ⊢ (;18 + 9) = ;27 |
15 | 1, 2, 3, 4, 14 | 4t3lem 12049 | 1 ⊢ (9 · 3) = ;27 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1525 (class class class)co 7023 1c1 10391 · cmul 10395 2c2 11546 3c3 11547 7c7 11551 8c8 11552 9c9 11553 ;cdc 11952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-ltxr 10533 df-sub 10725 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-dec 11953 |
This theorem is referenced by: 9t4e36 12076 3exp3 16258 prmlem2 16286 631prm 16293 1259lem4 16300 2503lem2 16304 4001lem3 16309 mcubic 25110 log2ublem3 25212 log2ub 25213 257prm 43227 fmtno4nprmfac193 43240 127prm 43267 m11nprm 43270 |
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