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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 11690 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 7156 | . 2 ⊢ (3 · 3) = (3 · (2 + 1)) |
3 | 3cn 11707 | . . . . 5 ⊢ 3 ∈ ℂ | |
4 | 2cn 11701 | . . . . 5 ⊢ 2 ∈ ℂ | |
5 | ax-1cn 10584 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | adddii 10642 | . . . 4 ⊢ (3 · (2 + 1)) = ((3 · 2) + (3 · 1)) |
7 | 3t2e6 11792 | . . . . 5 ⊢ (3 · 2) = 6 | |
8 | 3t1e3 11791 | . . . . 5 ⊢ (3 · 1) = 3 | |
9 | 7, 8 | oveq12i 7157 | . . . 4 ⊢ ((3 · 2) + (3 · 1)) = (6 + 3) |
10 | 6, 9 | eqtri 2844 | . . 3 ⊢ (3 · (2 + 1)) = (6 + 3) |
11 | 6p3e9 11786 | . . 3 ⊢ (6 + 3) = 9 | |
12 | 10, 11 | eqtri 2844 | . 2 ⊢ (3 · (2 + 1)) = 9 |
13 | 2, 12 | eqtri 2844 | 1 ⊢ (3 · 3) = 9 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 (class class class)co 7145 1c1 10527 + caddc 10529 · cmul 10531 2c2 11681 3c3 11682 6c6 11685 9c9 11688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-1rid 10596 ax-cnre 10599 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7148 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 |
This theorem is referenced by: sq3 13551 3dvds 15670 3dvdsdec 15671 3dvds2dec 15672 9nprm 16436 11prm 16438 43prm 16445 83prm 16446 317prm 16449 1259lem2 16455 1259lem4 16457 1259prm 16459 2503lem2 16461 mcubic 25352 log2tlbnd 25451 log2ublem3 25454 log2ub 25455 bposlem9 25796 lgsdir2lem5 25833 ex-lcm 28165 hgt750lem 31822 hgt750lem2 31823 3cubeslem3l 39163 3cubeslem3r 39164 inductionexd 40385 fmtno5lem3 43564 fmtno4prmfac193 43582 fmtno4nprmfac193 43583 127prm 43610 2exp340mod341 43745 9fppr8 43749 |
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