![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 8p3e11 | Structured version Visualization version GIF version |
Description: 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
8p3e11 | ⊢ (8 + 3) = ;11 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8nn0 11757 | . 2 ⊢ 8 ∈ ℕ0 | |
2 | 2nn0 11751 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | 0nn0 11749 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | df-3 11538 | . 2 ⊢ 3 = (2 + 1) | |
5 | 1e0p1 11978 | . 2 ⊢ 1 = (0 + 1) | |
6 | 8p2e10 12017 | . 2 ⊢ (8 + 2) = ;10 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 12007 | 1 ⊢ (8 + 3) = ;11 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1520 (class class class)co 7007 0cc0 10372 1c1 10373 + caddc 10375 2c2 11529 3c3 11530 8c8 11535 ;cdc 11936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-ov 7010 df-om 7428 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-ltxr 10515 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-dec 11937 |
This theorem is referenced by: 8p4e12 12019 317prm 16276 631prm 16277 1259lem2 16282 1259lem5 16285 2503lem1 16287 2503lem2 16288 4001lem1 16291 4001lem4 16294 fmtno5lem1 43151 2exp11 43201 nnsum4primesevenALTV 43402 |
Copyright terms: Public domain | W3C validator |