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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 11695 | . 2 ⊢ ;10 = (((9 + 1) · 1) + 0) | |
2 | 9nn 11393 | . . . . . 6 ⊢ 9 ∈ ℕ | |
3 | 1nn 11232 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | nnaddcl 11243 | . . . . . 6 ⊢ ((9 ∈ ℕ ∧ 1 ∈ ℕ) → (9 + 1) ∈ ℕ) | |
5 | 2, 3, 4 | mp2an 664 | . . . . 5 ⊢ (9 + 1) ∈ ℕ |
6 | 5 | nncni 11231 | . . . 4 ⊢ (9 + 1) ∈ ℂ |
7 | 6 | mulid1i 10243 | . . 3 ⊢ ((9 + 1) · 1) = (9 + 1) |
8 | 7 | oveq1i 6802 | . 2 ⊢ (((9 + 1) · 1) + 0) = ((9 + 1) + 0) |
9 | 6 | addid1i 10424 | . 2 ⊢ ((9 + 1) + 0) = (9 + 1) |
10 | 1, 8, 9 | 3eqtrri 2798 | 1 ⊢ (9 + 1) = ;10 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∈ wcel 2145 (class class class)co 6792 0cc0 10137 1c1 10138 + caddc 10140 · cmul 10142 ℕcn 11221 9c9 11278 ;cdc 11694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6795 df-om 7212 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-ltxr 10280 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-dec 11695 |
This theorem is referenced by: dfdec10 11698 10nn 11715 le9lt10 11730 decsucc 11751 5p5e10 11796 6p4e10 11798 7p3e10 11803 8p2e10 11810 9p2e11 11819 10m1e9 11830 9lt10 11873 sq10e99m1 13255 3dvds 15260 3dvdsdec 15262 3dvds2dec 15264 1259lem2 16045 1259lem3 16046 1259lem4 16047 2503lem2 16051 4001lem1 16054 4001lem2 16055 4001lem4 16057 bposlem4 25232 bposlem5 25233 dp2lt10 29928 1mhdrd 29961 hgt750lem2 31067 rmydioph 38103 127prm 42039 bgoldbtbndlem1 42217 |
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