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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 9009 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 5907 | . 2 ⊢ (3 · 3) = (3 · (2 + 1)) |
3 | 3cn 9024 | . . . . 5 ⊢ 3 ∈ ℂ | |
4 | 2cn 9020 | . . . . 5 ⊢ 2 ∈ ℂ | |
5 | ax-1cn 7934 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | adddii 7997 | . . . 4 ⊢ (3 · (2 + 1)) = ((3 · 2) + (3 · 1)) |
7 | 3t2e6 9105 | . . . . 5 ⊢ (3 · 2) = 6 | |
8 | 3t1e3 9104 | . . . . 5 ⊢ (3 · 1) = 3 | |
9 | 7, 8 | oveq12i 5908 | . . . 4 ⊢ ((3 · 2) + (3 · 1)) = (6 + 3) |
10 | 6, 9 | eqtri 2210 | . . 3 ⊢ (3 · (2 + 1)) = (6 + 3) |
11 | 6p3e9 9099 | . . 3 ⊢ (6 + 3) = 9 | |
12 | 10, 11 | eqtri 2210 | . 2 ⊢ (3 · (2 + 1)) = 9 |
13 | 2, 12 | eqtri 2210 | 1 ⊢ (3 · 3) = 9 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 (class class class)co 5896 1c1 7842 + caddc 7844 · cmul 7846 2c2 9000 3c3 9001 6c6 9004 9c9 9007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-1rid 7948 ax-cnre 7952 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5899 df-2 9008 df-3 9009 df-4 9010 df-5 9011 df-6 9012 df-7 9013 df-8 9014 df-9 9015 |
This theorem is referenced by: sq3 10648 3dvdsdec 11902 3dvds2dec 11903 lgsdir2lem5 14891 |
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