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Mirrors > Home > MPE Home > Th. List > 9p1e10 | Structured version Visualization version GIF version |
Description: 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.) |
Ref | Expression |
---|---|
9p1e10 | ⊢ (9 + 1) = ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dec 12577 | . 2 ⊢ ;10 = (((9 + 1) · 1) + 0) | |
2 | 9nn 12209 | . . . . . 6 ⊢ 9 ∈ ℕ | |
3 | 1nn 12122 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | nnaddcl 12134 | . . . . . 6 ⊢ ((9 ∈ ℕ ∧ 1 ∈ ℕ) → (9 + 1) ∈ ℕ) | |
5 | 2, 3, 4 | mp2an 690 | . . . . 5 ⊢ (9 + 1) ∈ ℕ |
6 | 5 | nncni 12121 | . . . 4 ⊢ (9 + 1) ∈ ℂ |
7 | 6 | mulid1i 11117 | . . 3 ⊢ ((9 + 1) · 1) = (9 + 1) |
8 | 7 | oveq1i 7361 | . 2 ⊢ (((9 + 1) · 1) + 0) = ((9 + 1) + 0) |
9 | 6 | addid1i 11300 | . 2 ⊢ ((9 + 1) + 0) = (9 + 1) |
10 | 1, 8, 9 | 3eqtrri 2770 | 1 ⊢ (9 + 1) = ;10 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 (class class class)co 7351 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 ℕcn 12111 9c9 12173 ;cdc 12576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-dec 12577 |
This theorem is referenced by: dfdec10 12579 10nn 12592 le9lt10 12603 decsucc 12617 5p5e10 12647 6p4e10 12648 7p3e10 12651 8p2e10 12656 9p2e11 12663 10m1e9 12672 9lt10 12707 sq10e99m1 14119 3dvds 16173 3dvdsdec 16174 3dvds2dec 16175 1259lem2 16964 1259lem3 16965 1259lem4 16966 2503lem2 16970 4001lem1 16973 4001lem2 16974 4001lem4 16976 bposlem4 26587 bposlem5 26588 dp2lt10 31566 1mhdrd 31598 hgt750lem2 33077 60gcd7e1 40400 lcmineqlem23 40446 sqdeccom12 40712 rmydioph 41247 127prm 45692 2exp340mod341 45826 bgoldbtbndlem1 45898 |
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