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| Mirrors > Home > MPE Home > Th. List > 6p4e10 | Structured version Visualization version GIF version | ||
| Description: 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
| Ref | Expression |
|---|---|
| 6p4e10 | ⊢ (6 + 4) = ;10 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-4 12331 | . . . 4 ⊢ 4 = (3 + 1) | |
| 2 | 1 | oveq2i 7442 | . . 3 ⊢ (6 + 4) = (6 + (3 + 1)) |
| 3 | 6cn 12357 | . . . 4 ⊢ 6 ∈ ℂ | |
| 4 | 3cn 12347 | . . . 4 ⊢ 3 ∈ ℂ | |
| 5 | ax-1cn 11213 | . . . 4 ⊢ 1 ∈ ℂ | |
| 6 | 3, 4, 5 | addassi 11271 | . . 3 ⊢ ((6 + 3) + 1) = (6 + (3 + 1)) |
| 7 | 2, 6 | eqtr4i 2768 | . 2 ⊢ (6 + 4) = ((6 + 3) + 1) |
| 8 | 6p3e9 12426 | . . 3 ⊢ (6 + 3) = 9 | |
| 9 | 8 | oveq1i 7441 | . 2 ⊢ ((6 + 3) + 1) = (9 + 1) |
| 10 | 9p1e10 12735 | . 2 ⊢ (9 + 1) = ;10 | |
| 11 | 7, 9, 10 | 3eqtri 2769 | 1 ⊢ (6 + 4) = ;10 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 3c3 12322 4c4 12323 6c6 12325 9c9 12328 ;cdc 12733 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-dec 12734 |
| This theorem is referenced by: 6p5e11 12806 6t5e30 12840 2exp11 17127 1259lem4 17171 1259lem5 17172 2503prm 17177 4001lem1 17178 4001prm 17182 log2ub 26992 ex-bc 30471 hgt750lem2 34667 420gcd8e4 42007 3lexlogpow5ineq1 42055 5bc2eq10 42143 fmtno5lem4 47543 fmtno5faclem2 47567 m11nprm 47588 |
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