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Mirrors > Home > ILE Home > Th. List > 3t3e9 | GIF version |
Description: 3 times 3 equals 9. (Contributed by NM, 11-May-2004.) |
Ref | Expression |
---|---|
3t3e9 | ⊢ (3 · 3) = 9 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3 8780 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 5785 | . 2 ⊢ (3 · 3) = (3 · (2 + 1)) |
3 | 3cn 8795 | . . . . 5 ⊢ 3 ∈ ℂ | |
4 | 2cn 8791 | . . . . 5 ⊢ 2 ∈ ℂ | |
5 | ax-1cn 7713 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | adddii 7776 | . . . 4 ⊢ (3 · (2 + 1)) = ((3 · 2) + (3 · 1)) |
7 | 3t2e6 8876 | . . . . 5 ⊢ (3 · 2) = 6 | |
8 | 3t1e3 8875 | . . . . 5 ⊢ (3 · 1) = 3 | |
9 | 7, 8 | oveq12i 5786 | . . . 4 ⊢ ((3 · 2) + (3 · 1)) = (6 + 3) |
10 | 6, 9 | eqtri 2160 | . . 3 ⊢ (3 · (2 + 1)) = (6 + 3) |
11 | 6p3e9 8870 | . . 3 ⊢ (6 + 3) = 9 | |
12 | 10, 11 | eqtri 2160 | . 2 ⊢ (3 · (2 + 1)) = 9 |
13 | 2, 12 | eqtri 2160 | 1 ⊢ (3 · 3) = 9 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 (class class class)co 5774 1c1 7621 + caddc 7623 · cmul 7625 2c2 8771 3c3 8772 6c6 8775 9c9 8778 |
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-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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-1rid 7727 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 |
This theorem is referenced by: sq3 10389 3dvdsdec 11562 3dvds2dec 11563 |
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