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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 12329 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | oveq2i 7442 | . . 3 ⊢ (5 + 4) = (5 + (3 + 1)) |
3 | 5cn 12352 | . . . 4 ⊢ 5 ∈ ℂ | |
4 | 3cn 12345 | . . . 4 ⊢ 3 ∈ ℂ | |
5 | ax-1cn 11211 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 11269 | . . 3 ⊢ ((5 + 3) + 1) = (5 + (3 + 1)) |
7 | 2, 6 | eqtr4i 2766 | . 2 ⊢ (5 + 4) = ((5 + 3) + 1) |
8 | df-9 12334 | . . 3 ⊢ 9 = (8 + 1) | |
9 | 5p3e8 12421 | . . . 4 ⊢ (5 + 3) = 8 | |
10 | 9 | oveq1i 7441 | . . 3 ⊢ ((5 + 3) + 1) = (8 + 1) |
11 | 8, 10 | eqtr4i 2766 | . 2 ⊢ 9 = ((5 + 3) + 1) |
12 | 7, 11 | eqtr4i 2766 | 1 ⊢ (5 + 4) = 9 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7431 1c1 11154 + caddc 11156 3c3 12320 4c4 12321 5c5 12322 8c8 12325 9c9 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-1cn 11211 ax-addcl 11213 ax-addass 11218 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 |
This theorem is referenced by: 5p5e10 12802 139prm 17158 1259lem3 17167 1259lem4 17168 2503lem2 17172 4001lem1 17175 4001lem2 17176 hgt750lem2 34646 problem1 35650 problem2 35651 resqrtvalex 43635 imsqrtvalex 43636 inductionexd 44145 139prmALT 47521 |
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