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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 10958 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | oveq2i 6560 | . . 3 ⊢ (5 + 4) = (5 + (3 + 1)) |
3 | 5cn 10977 | . . . 4 ⊢ 5 ∈ ℂ | |
4 | 3cn 10972 | . . . 4 ⊢ 3 ∈ ℂ | |
5 | ax-1cn 9873 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 9927 | . . 3 ⊢ ((5 + 3) + 1) = (5 + (3 + 1)) |
7 | 2, 6 | eqtr4i 2635 | . 2 ⊢ (5 + 4) = ((5 + 3) + 1) |
8 | df-9 10963 | . . 3 ⊢ 9 = (8 + 1) | |
9 | 5p3e8 11043 | . . . 4 ⊢ (5 + 3) = 8 | |
10 | 9 | oveq1i 6559 | . . 3 ⊢ ((5 + 3) + 1) = (8 + 1) |
11 | 8, 10 | eqtr4i 2635 | . 2 ⊢ 9 = ((5 + 3) + 1) |
12 | 7, 11 | eqtr4i 2635 | 1 ⊢ (5 + 4) = 9 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 (class class class)co 6549 1c1 9816 + caddc 9818 3c3 10948 4c4 10949 5c5 10950 8c8 10953 9c9 10954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-addass 9880 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 |
This theorem is referenced by: 5p5e10OLD 11045 5p5e10 11472 139prm 15669 1259lem3 15678 1259lem4 15679 2503lem2 15683 4001lem1 15686 4001lem2 15687 problem1 30812 problem2 30813 problem2OLD 30814 inductionexd 37473 139prmALT 40049 |
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