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Mirrors > Home > MPE Home > Th. List > 7p5e12 | Structured version Visualization version GIF version |
Description: 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
7p5e12 | ⊢ (7 + 5) = ;12 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7nn0 11600 | . 2 ⊢ 7 ∈ ℕ0 | |
2 | 4nn0 11597 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | 1nn0 11594 | . 2 ⊢ 1 ∈ ℕ0 | |
4 | df-5 11375 | . 2 ⊢ 5 = (4 + 1) | |
5 | df-2 11372 | . 2 ⊢ 2 = (1 + 1) | |
6 | 7p4e11 11857 | . 2 ⊢ (7 + 4) = ;11 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 11851 | 1 ⊢ (7 + 5) = ;12 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 (class class class)co 6876 1c1 10223 + caddc 10225 2c2 11364 4c4 11366 5c5 11367 7c7 11369 ;cdc 11779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-ov 6879 df-om 7298 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-ltxr 10366 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-dec 11780 |
This theorem is referenced by: 7p6e13 11859 7t6e42 11894 631prm 16158 1259lem3 16164 1259lem4 16165 2503lem1 16168 2503lem2 16169 2503lem3 16170 4001lem1 16172 127prm 42285 |
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