9p9e18 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 9p9e18		Structured version Visualization version GIF version

Theorem 9p9e18 12801

Description: 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)

**Assertion**
Ref	Expression
9p9e18	⊢ (9 + 9) = ;18

**Proof of Theorem 9p9e18**
Step	Hyp	Ref	Expression
1		9nn0 12526	. 2 ⊢ 9 ∈ ℕ₀
2		8nn0 12525	. 2 ⊢ 8 ∈ ℕ₀
3		7nn0 12524	. 2 ⊢ 7 ∈ ℕ₀
4		df-9 12312	. 2 ⊢ 9 = (8 + 1)
5		df-8 12311	. 2 ⊢ 8 = (7 + 1)
6		9p8e17 12800	. 2 ⊢ (9 + 8) = ;17
7	1, 2, 3, 4, 5, 6	6p5lem 12777	1 ⊢ (9 + 9) = ;18

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 (class class class)co 7420 1c1 11139 + caddc 11141 7c7 12302 8c8 12303 9c9 12304 cdc 12707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-dec 12708

This theorem is referenced by: 9t2e18 12829 prmlem2 17088 2503lem2 17106 2503lem3 17107

	
		OSZAR »