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| Mirrors > Home > MPE Home > Th. List > 3p3e6 | Structured version Visualization version GIF version | ||
| Description: 3 + 3 = 6. (Contributed by NM, 11-May-2004.) |
| Ref | Expression |
|---|---|
| 3p3e6 | ⊢ (3 + 3) = 6 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-3 12330 | . . . 4 ⊢ 3 = (2 + 1) | |
| 2 | 1 | oveq2i 7442 | . . 3 ⊢ (3 + 3) = (3 + (2 + 1)) |
| 3 | 3cn 12347 | . . . 4 ⊢ 3 ∈ ℂ | |
| 4 | 2cn 12341 | . . . 4 ⊢ 2 ∈ ℂ | |
| 5 | ax-1cn 11213 | . . . 4 ⊢ 1 ∈ ℂ | |
| 6 | 3, 4, 5 | addassi 11271 | . . 3 ⊢ ((3 + 2) + 1) = (3 + (2 + 1)) |
| 7 | 2, 6 | eqtr4i 2768 | . 2 ⊢ (3 + 3) = ((3 + 2) + 1) |
| 8 | df-6 12333 | . . 3 ⊢ 6 = (5 + 1) | |
| 9 | 3p2e5 12417 | . . . 4 ⊢ (3 + 2) = 5 | |
| 10 | 9 | oveq1i 7441 | . . 3 ⊢ ((3 + 2) + 1) = (5 + 1) |
| 11 | 8, 10 | eqtr4i 2768 | . 2 ⊢ 6 = ((3 + 2) + 1) |
| 12 | 7, 11 | eqtr4i 2768 | 1 ⊢ (3 + 3) = 6 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7431 1c1 11156 + caddc 11158 2c2 12321 3c3 12322 5c5 12324 6c6 12325 |
| 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-1cn 11213 ax-addcl 11215 ax-addass 11220 |
| 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-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 |
| This theorem is referenced by: 3t2e6 12432 163prm 17162 631prm 17164 2503prm 17177 binom4 26893 ex-dvds 30475 ex-gcd 30476 kur14lem8 35218 ex-decpmul 42340 3cubeslem3l 42697 gbegt5 47748 gboge9 47751 gbpart6 47753 gbpart9 47756 gbpart11 47757 zlmodzxzequa 48413 ackval3012 48613 ackval41a 48615 |
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