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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 7974 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 5523 | . 2 ⊢ (3 · 3) = (3 · (2 + 1)) |
3 | 3cn 7990 | . . . . 5 ⊢ 3 ∈ ℂ | |
4 | 2cn 7986 | . . . . 5 ⊢ 2 ∈ ℂ | |
5 | ax-1cn 6977 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | adddii 7037 | . . . 4 ⊢ (3 · (2 + 1)) = ((3 · 2) + (3 · 1)) |
7 | 3t2e6 8071 | . . . . 5 ⊢ (3 · 2) = 6 | |
8 | 3t1e3 8070 | . . . . 5 ⊢ (3 · 1) = 3 | |
9 | 7, 8 | oveq12i 5524 | . . . 4 ⊢ ((3 · 2) + (3 · 1)) = (6 + 3) |
10 | 6, 9 | eqtri 2060 | . . 3 ⊢ (3 · (2 + 1)) = (6 + 3) |
11 | 6p3e9 8062 | . . 3 ⊢ (6 + 3) = 9 | |
12 | 10, 11 | eqtri 2060 | . 2 ⊢ (3 · (2 + 1)) = 9 |
13 | 2, 12 | eqtri 2060 | 1 ⊢ (3 · 3) = 9 |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 (class class class)co 5512 1c1 6890 + caddc 6892 · cmul 6894 2c2 7964 3c3 7965 6c6 7968 9c9 7971 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-1rid 6991 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 df-2 7973 df-3 7974 df-4 7975 df-5 7976 df-6 7977 df-7 7978 df-8 7979 df-9 7980 |
This theorem is referenced by: sq3 9350 |
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