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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 8895 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | oveq2i 5836 | . . 3 ⊢ (5 + 4) = (5 + (3 + 1)) |
3 | 5cn 8914 | . . . 4 ⊢ 5 ∈ ℂ | |
4 | 3cn 8909 | . . . 4 ⊢ 3 ∈ ℂ | |
5 | ax-1cn 7826 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 7887 | . . 3 ⊢ ((5 + 3) + 1) = (5 + (3 + 1)) |
7 | 2, 6 | eqtr4i 2181 | . 2 ⊢ (5 + 4) = ((5 + 3) + 1) |
8 | df-9 8900 | . . 3 ⊢ 9 = (8 + 1) | |
9 | 5p3e8 8981 | . . . 4 ⊢ (5 + 3) = 8 | |
10 | 9 | oveq1i 5835 | . . 3 ⊢ ((5 + 3) + 1) = (8 + 1) |
11 | 8, 10 | eqtr4i 2181 | . 2 ⊢ 9 = ((5 + 3) + 1) |
12 | 7, 11 | eqtr4i 2181 | 1 ⊢ (5 + 4) = 9 |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 (class class class)co 5825 1c1 7734 + caddc 7736 3c3 8886 4c4 8887 5c5 8888 8c8 8891 9c9 8892 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-addrcl 7830 ax-addass 7835 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-iota 5136 df-fv 5179 df-ov 5828 df-2 8893 df-3 8894 df-4 8895 df-5 8896 df-6 8897 df-7 8898 df-8 8899 df-9 8900 |
This theorem is referenced by: 5p5e10 9366 |
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