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| 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 12330 | . . 3 ⊢ 3 = (2 + 1) | |
| 2 | 1 | oveq2i 7442 | . 2 ⊢ (3 · 3) = (3 · (2 + 1)) |
| 3 | 3cn 12347 | . . . . 5 ⊢ 3 ∈ ℂ | |
| 4 | 2cn 12341 | . . . . 5 ⊢ 2 ∈ ℂ | |
| 5 | ax-1cn 11213 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 6 | 3, 4, 5 | adddii 11273 | . . . 4 ⊢ (3 · (2 + 1)) = ((3 · 2) + (3 · 1)) |
| 7 | 3t2e6 12432 | . . . . 5 ⊢ (3 · 2) = 6 | |
| 8 | 3t1e3 12431 | . . . . 5 ⊢ (3 · 1) = 3 | |
| 9 | 7, 8 | oveq12i 7443 | . . . 4 ⊢ ((3 · 2) + (3 · 1)) = (6 + 3) |
| 10 | 6, 9 | eqtri 2765 | . . 3 ⊢ (3 · (2 + 1)) = (6 + 3) |
| 11 | 6p3e9 12426 | . . 3 ⊢ (6 + 3) = 9 | |
| 12 | 10, 11 | eqtri 2765 | . 2 ⊢ (3 · (2 + 1)) = 9 |
| 13 | 2, 12 | eqtri 2765 | 1 ⊢ (3 · 3) = 9 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7431 1c1 11156 + caddc 11158 · cmul 11160 2c2 12321 3c3 12322 6c6 12325 9c9 12328 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-1rid 11225 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 |
| This theorem is referenced by: sq3 14237 3dvds 16368 3dvdsdec 16369 3dvds2dec 16370 9nprm 17150 11prm 17152 43prm 17159 83prm 17160 317prm 17163 1259lem2 17169 1259lem4 17171 1259prm 17173 2503lem2 17175 mcubic 26890 log2tlbnd 26988 log2ublem3 26991 log2ub 26992 bposlem9 27336 lgsdir2lem5 27373 ex-lcm 30477 hgt750lem 34666 hgt750lem2 34667 3lexlogpow2ineq2 42060 3lexlogpow5ineq5 42061 3cubeslem3l 42697 3cubeslem3r 42698 inductionexd 44168 fmtno5lem3 47542 fmtno4prmfac193 47560 fmtno4nprmfac193 47561 127prm 47586 2exp340mod341 47720 9fppr8 47724 |
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