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| Mirrors > Home > MPE Home > Th. List > 5p4e9 | Structured version Visualization version GIF version | ||
| Description: 5 + 4 = 9. (Contributed by NM, 11-May-2004.) | 
| Ref | Expression | 
|---|---|
| 5p4e9 | ⊢ (5 + 4) = 9 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-4 12332 | . . . 4 ⊢ 4 = (3 + 1) | |
| 2 | 1 | oveq2i 7443 | . . 3 ⊢ (5 + 4) = (5 + (3 + 1)) | 
| 3 | 5cn 12355 | . . . 4 ⊢ 5 ∈ ℂ | |
| 4 | 3cn 12348 | . . . 4 ⊢ 3 ∈ ℂ | |
| 5 | ax-1cn 11214 | . . . 4 ⊢ 1 ∈ ℂ | |
| 6 | 3, 4, 5 | addassi 11272 | . . 3 ⊢ ((5 + 3) + 1) = (5 + (3 + 1)) | 
| 7 | 2, 6 | eqtr4i 2767 | . 2 ⊢ (5 + 4) = ((5 + 3) + 1) | 
| 8 | df-9 12337 | . . 3 ⊢ 9 = (8 + 1) | |
| 9 | 5p3e8 12424 | . . . 4 ⊢ (5 + 3) = 8 | |
| 10 | 9 | oveq1i 7442 | . . 3 ⊢ ((5 + 3) + 1) = (8 + 1) | 
| 11 | 8, 10 | eqtr4i 2767 | . 2 ⊢ 9 = ((5 + 3) + 1) | 
| 12 | 7, 11 | eqtr4i 2767 | 1 ⊢ (5 + 4) = 9 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 (class class class)co 7432 1c1 11157 + caddc 11159 3c3 12323 4c4 12324 5c5 12325 8c8 12328 9c9 12329 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-1cn 11214 ax-addcl 11216 ax-addass 11221 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 | 
| This theorem is referenced by: 5p5e10 12806 139prm 17162 1259lem3 17171 1259lem4 17172 2503lem2 17176 4001lem1 17179 4001lem2 17180 hgt750lem2 34668 problem1 35671 problem2 35672 resqrtvalex 43663 imsqrtvalex 43664 inductionexd 44173 139prmALT 47588 | 
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