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Mirrors > Home > MPE Home > Th. List > 5p5e10 | Structured version Visualization version GIF version |
Description: 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
5p5e10 | ⊢ (5 + 5) = ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 11504 | . . . 4 ⊢ 5 = (4 + 1) | |
2 | 1 | oveq2i 6985 | . . 3 ⊢ (5 + 5) = (5 + (4 + 1)) |
3 | 5cn 11528 | . . . 4 ⊢ 5 ∈ ℂ | |
4 | 4cn 11524 | . . . 4 ⊢ 4 ∈ ℂ | |
5 | ax-1cn 10391 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | 3, 4, 5 | addassi 10448 | . . 3 ⊢ ((5 + 4) + 1) = (5 + (4 + 1)) |
7 | 2, 6 | eqtr4i 2798 | . 2 ⊢ (5 + 5) = ((5 + 4) + 1) |
8 | 5p4e9 11603 | . . 3 ⊢ (5 + 4) = 9 | |
9 | 8 | oveq1i 6984 | . 2 ⊢ ((5 + 4) + 1) = (9 + 1) |
10 | 9p1e10 11911 | . 2 ⊢ (9 + 1) = ;10 | |
11 | 7, 9, 10 | 3eqtri 2799 | 1 ⊢ (5 + 5) = ;10 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 (class class class)co 6974 0cc0 10333 1c1 10334 + caddc 10336 4c4 11495 5c5 11496 9c9 11500 ;cdc 11909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-ltxr 10477 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-dec 11910 |
This theorem is referenced by: 5t2e10 12011 5t4e20 12013 2503lem2 16325 log2ublem3 25243 threehalves 30361 hgt750lem2 31603 sqn5i 38641 235t711 38647 bgoldbtbndlem1 43372 |
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