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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 12329 | . . . 4 ⊢ 4 = (3 + 1) | |
2 | 1 | oveq2i 7442 | . . 3 ⊢ (6 + 4) = (6 + (3 + 1)) |
3 | 6cn 12355 | . . . 4 ⊢ 6 ∈ ℂ | |
4 | 3cn 12345 | . . . 4 ⊢ 3 ∈ ℂ | |
5 | ax-1cn 11211 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 11269 | . . 3 ⊢ ((6 + 3) + 1) = (6 + (3 + 1)) |
7 | 2, 6 | eqtr4i 2766 | . 2 ⊢ (6 + 4) = ((6 + 3) + 1) |
8 | 6p3e9 12424 | . . 3 ⊢ (6 + 3) = 9 | |
9 | 8 | oveq1i 7441 | . 2 ⊢ ((6 + 3) + 1) = (9 + 1) |
10 | 9p1e10 12733 | . 2 ⊢ (9 + 1) = ;10 | |
11 | 7, 9, 10 | 3eqtri 2767 | 1 ⊢ (6 + 4) = ;10 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 3c3 12320 4c4 12321 6c6 12323 9c9 12326 ;cdc 12731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-dec 12732 |
This theorem is referenced by: 6p5e11 12804 6t5e30 12838 2exp11 17124 1259lem4 17168 1259lem5 17169 2503prm 17174 4001lem1 17175 4001prm 17179 log2ub 27007 ex-bc 30481 hgt750lem2 34646 420gcd8e4 41988 3lexlogpow5ineq1 42036 5bc2eq10 42124 fmtno5lem4 47481 fmtno5faclem2 47505 m11nprm 47526 |
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