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Mirrors > Home > MPE Home > Th. List > 9p8e17 | Structured version Visualization version GIF version |
Description: 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9p8e17 | ⊢ (9 + 8) = ;17 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 12483 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 7nn0 12481 | . 2 ⊢ 7 ∈ ℕ0 | |
3 | 6nn0 12480 | . 2 ⊢ 6 ∈ ℕ0 | |
4 | df-8 12268 | . 2 ⊢ 8 = (7 + 1) | |
5 | df-7 12267 | . 2 ⊢ 7 = (6 + 1) | |
6 | 9p7e16 12756 | . 2 ⊢ (9 + 7) = ;16 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 12734 | 1 ⊢ (9 + 8) = ;17 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7396 1c1 11098 + caddc 11100 6c6 12258 7c7 12259 8c8 12260 9c9 12261 ;cdc 12664 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-ltxr 11240 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-dec 12665 |
This theorem is referenced by: 9p9e18 12758 9t3e27 12787 37prm 17041 317prm 17046 2503lem2 17058 2503lem3 17059 fmtno4nprmfac193 46115 127prm 46140 |
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