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| Mirrors > Home > MPE Home > Th. List > 4p3e7 | Structured version Visualization version GIF version | ||
| Description: 4 + 3 = 7. (Contributed by NM, 11-May-2004.) |
| Ref | Expression |
|---|---|
| 4p3e7 | ⊢ (4 + 3) = 7 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-3 12330 | . . . 4 ⊢ 3 = (2 + 1) | |
| 2 | 1 | oveq2i 7442 | . . 3 ⊢ (4 + 3) = (4 + (2 + 1)) |
| 3 | 4cn 12351 | . . . 4 ⊢ 4 ∈ ℂ | |
| 4 | 2cn 12341 | . . . 4 ⊢ 2 ∈ ℂ | |
| 5 | ax-1cn 11213 | . . . 4 ⊢ 1 ∈ ℂ | |
| 6 | 3, 4, 5 | addassi 11271 | . . 3 ⊢ ((4 + 2) + 1) = (4 + (2 + 1)) |
| 7 | 2, 6 | eqtr4i 2768 | . 2 ⊢ (4 + 3) = ((4 + 2) + 1) |
| 8 | df-7 12334 | . . 3 ⊢ 7 = (6 + 1) | |
| 9 | 4p2e6 12419 | . . . 4 ⊢ (4 + 2) = 6 | |
| 10 | 9 | oveq1i 7441 | . . 3 ⊢ ((4 + 2) + 1) = (6 + 1) |
| 11 | 8, 10 | eqtr4i 2768 | . 2 ⊢ 7 = ((4 + 2) + 1) |
| 12 | 7, 11 | eqtr4i 2768 | 1 ⊢ (4 + 3) = 7 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7431 1c1 11156 + caddc 11158 2c2 12321 3c3 12322 4c4 12323 6c6 12325 7c7 12326 |
| 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 df-7 12334 |
| This theorem is referenced by: 4p4e8 12421 hash7g 14525 37prm 17158 317prm 17163 1259lem5 17172 2503lem2 17175 4001lem1 17178 4001lem2 17179 log2ub 26992 bposlem8 27335 2lgslem3d 27443 2lgsoddprmlem3d 27457 hgt750lem 34666 hgt750lem2 34667 fmtno5lem4 47543 257prm 47548 127prm 47586 gbpart7 47754 sbgoldbwt 47764 sbgoldbst 47765 ackval2012 48612 |
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