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Mirrors > Home > ILE Home > Th. List > 9t3e27 | 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 9001 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 2nn0 8994 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 8780 | . 2 ⊢ 3 = (2 + 1) | |
4 | 9t2e18 9303 | . 2 ⊢ (9 · 2) = ;18 | |
5 | 1nn0 8993 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 8nn0 9000 | . . 3 ⊢ 8 ∈ ℕ0 | |
7 | eqid 2139 | . . 3 ⊢ ;18 = ;18 | |
8 | 1p1e2 8837 | . . 3 ⊢ (1 + 1) = 2 | |
9 | 7nn0 8999 | . . 3 ⊢ 7 ∈ ℕ0 | |
10 | 1 | nn0cni 8989 | . . . 4 ⊢ 9 ∈ ℂ |
11 | 6 | nn0cni 8989 | . . . 4 ⊢ 8 ∈ ℂ |
12 | 9p8e17 9274 | . . . 4 ⊢ (9 + 8) = ;17 | |
13 | 10, 11, 12 | addcomli 7907 | . . 3 ⊢ (8 + 9) = ;17 |
14 | 5, 6, 1, 7, 8, 9, 13 | decaddci 9242 | . 2 ⊢ (;18 + 9) = ;27 |
15 | 1, 2, 3, 4, 14 | 4t3lem 9278 | 1 ⊢ (9 · 3) = ;27 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 (class class class)co 5774 1c1 7621 · cmul 7625 2c2 8771 3c3 8772 7c7 8776 8c8 8777 9c9 8778 ;cdc 9182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 df-n0 8978 df-dec 9183 |
This theorem is referenced by: 9t4e36 9305 |
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