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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 11445 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | oveq2i 6935 | . . 3 ⊢ (5 + 4) = (5 + (3 + 1)) |
3 | 5cn 11470 | . . . 4 ⊢ 5 ∈ ℂ | |
4 | 3cn 11461 | . . . 4 ⊢ 3 ∈ ℂ | |
5 | ax-1cn 10332 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 10389 | . . 3 ⊢ ((5 + 3) + 1) = (5 + (3 + 1)) |
7 | 2, 6 | eqtr4i 2805 | . 2 ⊢ (5 + 4) = ((5 + 3) + 1) |
8 | df-9 11450 | . . 3 ⊢ 9 = (8 + 1) | |
9 | 5p3e8 11544 | . . . 4 ⊢ (5 + 3) = 8 | |
10 | 9 | oveq1i 6934 | . . 3 ⊢ ((5 + 3) + 1) = (8 + 1) |
11 | 8, 10 | eqtr4i 2805 | . 2 ⊢ 9 = ((5 + 3) + 1) |
12 | 7, 11 | eqtr4i 2805 | 1 ⊢ (5 + 4) = 9 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 (class class class)co 6924 1c1 10275 + caddc 10277 3c3 11436 4c4 11437 5c5 11438 8c8 11441 9c9 11442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-1cn 10332 ax-addcl 10334 ax-addass 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-iota 6101 df-fv 6145 df-ov 6927 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 |
This theorem is referenced by: 5p5e10 11923 139prm 16240 1259lem3 16249 1259lem4 16250 2503lem2 16254 4001lem1 16257 4001lem2 16258 hgt750lem2 31340 problem1 32164 problem2 32165 inductionexd 39423 139prmALT 42546 |
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