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Mirrors > Home > ILE Home > Th. List > 5p4e9 | 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 8774 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | oveq2i 5778 | . . 3 ⊢ (5 + 4) = (5 + (3 + 1)) |
3 | 5cn 8793 | . . . 4 ⊢ 5 ∈ ℂ | |
4 | 3cn 8788 | . . . 4 ⊢ 3 ∈ ℂ | |
5 | ax-1cn 7706 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 7767 | . . 3 ⊢ ((5 + 3) + 1) = (5 + (3 + 1)) |
7 | 2, 6 | eqtr4i 2161 | . 2 ⊢ (5 + 4) = ((5 + 3) + 1) |
8 | df-9 8779 | . . 3 ⊢ 9 = (8 + 1) | |
9 | 5p3e8 8860 | . . . 4 ⊢ (5 + 3) = 8 | |
10 | 9 | oveq1i 5777 | . . 3 ⊢ ((5 + 3) + 1) = (8 + 1) |
11 | 8, 10 | eqtr4i 2161 | . 2 ⊢ 9 = ((5 + 3) + 1) |
12 | 7, 11 | eqtr4i 2161 | 1 ⊢ (5 + 4) = 9 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 (class class class)co 5767 1c1 7614 + caddc 7616 3c3 8765 4c4 8766 5c5 8767 8c8 8770 9c9 8771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-addrcl 7710 ax-addass 7715 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-7 8777 df-8 8778 df-9 8779 |
This theorem is referenced by: 5p5e10 9245 |
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