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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 12615 | . 2 ⊢ ;10 = (((9 + 1) · 1) + 0) | |
2 | 9nn 12247 | . . . . . 6 ⊢ 9 ∈ ℕ | |
3 | 1nn 12160 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | nnaddcl 12172 | . . . . . 6 ⊢ ((9 ∈ ℕ ∧ 1 ∈ ℕ) → (9 + 1) ∈ ℕ) | |
5 | 2, 3, 4 | mp2an 690 | . . . . 5 ⊢ (9 + 1) ∈ ℕ |
6 | 5 | nncni 12159 | . . . 4 ⊢ (9 + 1) ∈ ℂ |
7 | 6 | mulid1i 11155 | . . 3 ⊢ ((9 + 1) · 1) = (9 + 1) |
8 | 7 | oveq1i 7363 | . 2 ⊢ (((9 + 1) · 1) + 0) = ((9 + 1) + 0) |
9 | 6 | addid1i 11338 | . 2 ⊢ ((9 + 1) + 0) = (9 + 1) |
10 | 1, 8, 9 | 3eqtrri 2769 | 1 ⊢ (9 + 1) = ;10 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 (class class class)co 7353 0cc0 11047 1c1 11048 + caddc 11050 · cmul 11052 ℕcn 12149 9c9 12211 ;cdc 12614 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 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 7356 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-ltxr 11190 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-dec 12615 |
This theorem is referenced by: dfdec10 12617 10nn 12630 le9lt10 12641 decsucc 12655 5p5e10 12685 6p4e10 12686 7p3e10 12689 8p2e10 12694 9p2e11 12701 10m1e9 12710 9lt10 12745 sq10e99m1 14157 3dvds 16205 3dvdsdec 16206 3dvds2dec 16207 1259lem2 16996 1259lem3 16997 1259lem4 16998 2503lem2 17002 4001lem1 17005 4001lem2 17006 4001lem4 17008 bposlem4 26619 bposlem5 26620 dp2lt10 31623 1mhdrd 31655 hgt750lem2 33134 60gcd7e1 40429 lcmineqlem23 40475 sqdeccom12 40741 rmydioph 41276 127prm 45723 2exp340mod341 45857 bgoldbtbndlem1 45929 |
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