Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 3t3e9 | Structured version Visualization version 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 10957 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 6560 | . 2 ⊢ (3 · 3) = (3 · (2 + 1)) |
3 | 3cn 10972 | . . . . 5 ⊢ 3 ∈ ℂ | |
4 | 2cn 10968 | . . . . 5 ⊢ 2 ∈ ℂ | |
5 | ax-1cn 9873 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | adddii 9929 | . . . 4 ⊢ (3 · (2 + 1)) = ((3 · 2) + (3 · 1)) |
7 | 3t2e6 11056 | . . . . 5 ⊢ (3 · 2) = 6 | |
8 | 3t1e3 11055 | . . . . 5 ⊢ (3 · 1) = 3 | |
9 | 7, 8 | oveq12i 6561 | . . . 4 ⊢ ((3 · 2) + (3 · 1)) = (6 + 3) |
10 | 6, 9 | eqtri 2632 | . . 3 ⊢ (3 · (2 + 1)) = (6 + 3) |
11 | 6p3e9 11047 | . . 3 ⊢ (6 + 3) = 9 | |
12 | 10, 11 | eqtri 2632 | . 2 ⊢ (3 · (2 + 1)) = 9 |
13 | 2, 12 | eqtri 2632 | 1 ⊢ (3 · 3) = 9 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 (class class class)co 6549 1c1 9816 + caddc 9818 · cmul 9820 2c2 10947 3c3 10948 6c6 10951 9c9 10954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 |
This theorem is referenced by: sq3 12823 3dvds 14890 3dvdsOLD 14891 3dvdsdec 14892 3dvdsdecOLD 14893 3dvds2dec 14894 3dvds2decOLD 14895 9nprm 15657 11prm 15660 43prm 15667 83prm 15668 317prm 15671 1259lem2 15677 1259lem4 15679 1259prm 15681 2503lem2 15683 mcubic 24374 log2tlbnd 24472 log2ublem3 24475 log2ub 24476 bposlem9 24817 lgsdir2lem5 24854 ex-lcm 26707 inductionexd 37473 fmtno5lem3 40005 fmtno4prmfac193 40023 fmtno4nprmfac193 40024 127prm 40053 |
Copyright terms: Public domain | W3C validator |