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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 11282 | . . . 4 ⊢ 3 = (2 + 1) | |
2 | 1 | oveq2i 6804 | . . 3 ⊢ (7 + 3) = (7 + (2 + 1)) |
3 | 7cn 11306 | . . . 4 ⊢ 7 ∈ ℂ | |
4 | 2cn 11293 | . . . 4 ⊢ 2 ∈ ℂ | |
5 | ax-1cn 10196 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 10250 | . . 3 ⊢ ((7 + 2) + 1) = (7 + (2 + 1)) |
7 | 2, 6 | eqtr4i 2796 | . 2 ⊢ (7 + 3) = ((7 + 2) + 1) |
8 | 7p2e9 11374 | . . 3 ⊢ (7 + 2) = 9 | |
9 | 8 | oveq1i 6803 | . 2 ⊢ ((7 + 2) + 1) = (9 + 1) |
10 | 9p1e10 11698 | . 2 ⊢ (9 + 1) = ;10 | |
11 | 7, 9, 10 | 3eqtri 2797 | 1 ⊢ (7 + 3) = ;10 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 (class class class)co 6793 0cc0 10138 1c1 10139 + caddc 10141 2c2 11272 3c3 11273 7c7 11277 9c9 11279 ;cdc 11695 |
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 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 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 6796 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-ltxr 10281 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-dec 11696 |
This theorem is referenced by: 7p4e11 11806 1259lem4 16048 2503lem2 16052 2503lem3 16053 4001lem4 16058 log2ublem3 24896 log2ub 24897 127prm 42043 evengpoap3 42215 |
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