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Mirrors > Home > MPE Home > Th. List > 8p5e13 | Structured version Visualization version GIF version |
Description: 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
8p5e13 | ⊢ (8 + 5) = ;13 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8nn0 12244 | . 2 ⊢ 8 ∈ ℕ0 | |
2 | 4nn0 12240 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | 2nn0 12238 | . 2 ⊢ 2 ∈ ℕ0 | |
4 | df-5 12027 | . 2 ⊢ 5 = (4 + 1) | |
5 | df-3 12025 | . 2 ⊢ 3 = (2 + 1) | |
6 | 8p4e12 12507 | . 2 ⊢ (8 + 4) = ;12 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 12495 | 1 ⊢ (8 + 5) = ;13 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 (class class class)co 7268 1c1 10860 + caddc 10862 2c2 12016 3c3 12017 4c4 12018 5c5 12019 8c8 12022 ;cdc 12425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-ov 7271 df-om 7704 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-pnf 10999 df-mnf 11000 df-ltxr 11002 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-dec 12426 |
This theorem is referenced by: 8p6e14 12509 2exp16 16780 43prm 16811 83prm 16812 139prm 16813 317prm 16815 631prm 16816 1259lem3 16822 hgt750lem2 32618 139prmALT 45004 |
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