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Mirrors > Home > MPE Home > Th. List > 8p2e10 | Structured version Visualization version GIF version |
Description: 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
8p2e10 | ⊢ (8 + 2) = ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 11687 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq2i 7153 | . . 3 ⊢ (8 + 2) = (8 + (1 + 1)) |
3 | 8cn 11721 | . . . 4 ⊢ 8 ∈ ℂ | |
4 | ax-1cn 10581 | . . . 4 ⊢ 1 ∈ ℂ | |
5 | 3, 4, 4 | addassi 10637 | . . 3 ⊢ ((8 + 1) + 1) = (8 + (1 + 1)) |
6 | 2, 5 | eqtr4i 2847 | . 2 ⊢ (8 + 2) = ((8 + 1) + 1) |
7 | df-9 11694 | . . 3 ⊢ 9 = (8 + 1) | |
8 | 7 | oveq1i 7152 | . 2 ⊢ (9 + 1) = ((8 + 1) + 1) |
9 | 9p1e10 12087 | . 2 ⊢ (9 + 1) = ;10 | |
10 | 6, 8, 9 | 3eqtr2i 2850 | 1 ⊢ (8 + 2) = ;10 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7142 0cc0 10523 1c1 10524 + caddc 10526 2c2 11679 8c8 11685 9c9 11686 ;cdc 12085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-om 7567 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-ltxr 10666 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-dec 12086 |
This theorem is referenced by: 8p3e11 12166 8t5e40 12203 1259lem3 16449 1259lem4 16450 2503lem2 16454 4001lem1 16457 4001lem3 16459 4001prm 16461 m11nprm 43851 |
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