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Mirrors > Home > MPE Home > Th. List > 5t3e15 | Structured version Visualization version GIF version |
Description: 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
5t3e15 | ⊢ (5 · 3) = ;15 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 11906 | . 2 ⊢ 5 ∈ ℕ0 | |
2 | 2nn0 11903 | . 2 ⊢ 2 ∈ ℕ0 | |
3 | df-3 11690 | . 2 ⊢ 3 = (2 + 1) | |
4 | 5t2e10 12187 | . 2 ⊢ (5 · 2) = ;10 | |
5 | dec10p 12130 | . 2 ⊢ (;10 + 5) = ;15 | |
6 | 1, 2, 3, 4, 5 | 4t3lem 12184 | 1 ⊢ (5 · 3) = ;15 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 (class class class)co 7145 0cc0 10526 1c1 10527 · cmul 10531 2c2 11681 3c3 11682 5c5 11684 ;cdc 12087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-dec 12088 |
This theorem is referenced by: 5t4e20 12189 17prm 16440 prmlem2 16443 163prm 16448 317prm 16449 1259lem4 16457 2503lem2 16461 4001prm 16468 log2ub 25455 hgt750lem2 31823 inductionexd 40385 fmtno5lem2 43563 fmtno5lem3 43564 fmtno4prmfac193 43582 127prm 43610 |
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