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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 11119 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | oveq2i 6701 | . . 3 ⊢ (5 + 4) = (5 + (3 + 1)) |
3 | 5cn 11138 | . . . 4 ⊢ 5 ∈ ℂ | |
4 | 3cn 11133 | . . . 4 ⊢ 3 ∈ ℂ | |
5 | ax-1cn 10032 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 10086 | . . 3 ⊢ ((5 + 3) + 1) = (5 + (3 + 1)) |
7 | 2, 6 | eqtr4i 2676 | . 2 ⊢ (5 + 4) = ((5 + 3) + 1) |
8 | df-9 11124 | . . 3 ⊢ 9 = (8 + 1) | |
9 | 5p3e8 11204 | . . . 4 ⊢ (5 + 3) = 8 | |
10 | 9 | oveq1i 6700 | . . 3 ⊢ ((5 + 3) + 1) = (8 + 1) |
11 | 8, 10 | eqtr4i 2676 | . 2 ⊢ 9 = ((5 + 3) + 1) |
12 | 7, 11 | eqtr4i 2676 | 1 ⊢ (5 + 4) = 9 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 (class class class)co 6690 1c1 9975 + caddc 9977 3c3 11109 4c4 11110 5c5 11111 8c8 11114 9c9 11115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-addass 10039 ax-i2m1 10042 ax-1ne0 10043 ax-rrecex 10046 ax-cnre 10047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-iota 5889 df-fv 5934 df-ov 6693 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 |
This theorem is referenced by: 5p5e10OLD 11206 5p5e10 11634 139prm 15878 1259lem3 15887 1259lem4 15888 2503lem2 15892 4001lem1 15895 4001lem2 15896 hgt750lem2 30858 problem1 31684 problem2 31685 problem2OLD 31686 inductionexd 38770 139prmALT 41836 |
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