![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12218 | . . . 4 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 7369 | . . 3 ⊢ (7 + 3) = (7 + (2 + 1)) |
3 | 7cn 12248 | . . . 4 ⊢ 7 ∈ ℂ | |
4 | 2cn 12229 | . . . 4 ⊢ 2 ∈ ℂ | |
5 | ax-1cn 11110 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 11166 | . . 3 ⊢ ((7 + 2) + 1) = (7 + (2 + 1)) |
7 | 2, 6 | eqtr4i 2768 | . 2 ⊢ (7 + 3) = ((7 + 2) + 1) |
8 | 7p2e9 12315 | . . 3 ⊢ (7 + 2) = 9 | |
9 | 8 | oveq1i 7368 | . 2 ⊢ ((7 + 2) + 1) = (9 + 1) |
10 | 9p1e10 12621 | . 2 ⊢ (9 + 1) = ;10 | |
11 | 7, 9, 10 | 3eqtri 2769 | 1 ⊢ (7 + 3) = ;10 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7358 0cc0 11052 1c1 11053 + caddc 11055 2c2 12209 3c3 12210 7c7 12214 9c9 12216 ;cdc 12619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-ltxr 11195 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-dec 12620 |
This theorem is referenced by: 7p4e11 12695 1259lem4 17007 2503lem2 17011 2503lem3 17012 4001lem4 17017 log2ublem3 26301 log2ub 26302 ex-decpmul 40809 127prm 45798 evengpoap3 45998 |
Copyright terms: Public domain | W3C validator |