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Mirrors > Home > MPE Home > Th. List > 7p3e10 | Structured version Visualization version GIF version |
Description: 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
7p3e10 | ⊢ (7 + 3) = ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3 11891 | . . . 4 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 7221 | . . 3 ⊢ (7 + 3) = (7 + (2 + 1)) |
3 | 7cn 11921 | . . . 4 ⊢ 7 ∈ ℂ | |
4 | 2cn 11902 | . . . 4 ⊢ 2 ∈ ℂ | |
5 | ax-1cn 10784 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 10840 | . . 3 ⊢ ((7 + 2) + 1) = (7 + (2 + 1)) |
7 | 2, 6 | eqtr4i 2768 | . 2 ⊢ (7 + 3) = ((7 + 2) + 1) |
8 | 7p2e9 11988 | . . 3 ⊢ (7 + 2) = 9 | |
9 | 8 | oveq1i 7220 | . 2 ⊢ ((7 + 2) + 1) = (9 + 1) |
10 | 9p1e10 12292 | . 2 ⊢ (9 + 1) = ;10 | |
11 | 7, 9, 10 | 3eqtri 2769 | 1 ⊢ (7 + 3) = ;10 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 (class class class)co 7210 0cc0 10726 1c1 10727 + caddc 10729 2c2 11882 3c3 11883 7c7 11887 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: 7p4e11 12366 1259lem4 16684 2503lem2 16688 2503lem3 16689 4001lem4 16694 log2ublem3 25828 log2ub 25829 ex-decpmul 40025 127prm 44722 evengpoap3 44922 |
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