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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 8938 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 5864 | . 2 ⊢ (3 · 3) = (3 · (2 + 1)) |
3 | 3cn 8953 | . . . . 5 ⊢ 3 ∈ ℂ | |
4 | 2cn 8949 | . . . . 5 ⊢ 2 ∈ ℂ | |
5 | ax-1cn 7867 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | adddii 7930 | . . . 4 ⊢ (3 · (2 + 1)) = ((3 · 2) + (3 · 1)) |
7 | 3t2e6 9034 | . . . . 5 ⊢ (3 · 2) = 6 | |
8 | 3t1e3 9033 | . . . . 5 ⊢ (3 · 1) = 3 | |
9 | 7, 8 | oveq12i 5865 | . . . 4 ⊢ ((3 · 2) + (3 · 1)) = (6 + 3) |
10 | 6, 9 | eqtri 2191 | . . 3 ⊢ (3 · (2 + 1)) = (6 + 3) |
11 | 6p3e9 9028 | . . 3 ⊢ (6 + 3) = 9 | |
12 | 10, 11 | eqtri 2191 | . 2 ⊢ (3 · (2 + 1)) = 9 |
13 | 2, 12 | eqtri 2191 | 1 ⊢ (3 · 3) = 9 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 (class class class)co 5853 1c1 7775 + caddc 7777 · cmul 7779 2c2 8929 3c3 8930 6c6 8933 9c9 8936 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-1rid 7881 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-9 8944 |
This theorem is referenced by: sq3 10572 3dvdsdec 11824 3dvds2dec 11825 lgsdir2lem5 13727 |
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