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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 12291 | . 2 ⊢ ;10 = (((9 + 1) · 1) + 0) | |
2 | 9nn 11925 | . . . . . 6 ⊢ 9 ∈ ℕ | |
3 | 1nn 11838 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | nnaddcl 11850 | . . . . . 6 ⊢ ((9 ∈ ℕ ∧ 1 ∈ ℕ) → (9 + 1) ∈ ℕ) | |
5 | 2, 3, 4 | mp2an 692 | . . . . 5 ⊢ (9 + 1) ∈ ℕ |
6 | 5 | nncni 11837 | . . . 4 ⊢ (9 + 1) ∈ ℂ |
7 | 6 | mulid1i 10834 | . . 3 ⊢ ((9 + 1) · 1) = (9 + 1) |
8 | 7 | oveq1i 7220 | . 2 ⊢ (((9 + 1) · 1) + 0) = ((9 + 1) + 0) |
9 | 6 | addid1i 11016 | . 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 1543 ∈ wcel 2110 (class class class)co 7210 0cc0 10726 1c1 10727 + caddc 10729 · cmul 10731 ℕcn 11827 9c9 11889 ;cdc 12290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-ov 7213 df-om 7642 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-ltxr 10869 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-dec 12291 |
This theorem is referenced by: dfdec10 12293 10nn 12306 le9lt10 12317 decsucc 12331 5p5e10 12361 6p4e10 12362 7p3e10 12365 8p2e10 12370 9p2e11 12377 10m1e9 12386 9lt10 12421 sq10e99m1 13828 3dvds 15889 3dvdsdec 15890 3dvds2dec 15891 1259lem2 16682 1259lem3 16683 1259lem4 16684 2503lem2 16688 4001lem1 16691 4001lem2 16692 4001lem4 16694 bposlem4 26165 bposlem5 26166 dp2lt10 30875 1mhdrd 30907 hgt750lem2 32341 60gcd7e1 39745 lcmineqlem23 39791 sqdeccom12 40022 rmydioph 40537 127prm 44722 2exp340mod341 44856 bgoldbtbndlem1 44928 |
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