7t3e21 - Metamath Proof Explorer

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Theorem 7t3e21 12811

Description: 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)

**Assertion**
Ref	Expression
7t3e21	⊢ (7 · 3) = ;21

**Proof of Theorem 7t3e21**
Step	Hyp	Ref	Expression
1		7nn0 12518	. 2 ⊢ 7 ∈ ℕ₀
2		2nn0 12513	. 2 ⊢ 2 ∈ ℕ₀
3		df-3 12300	. 2 ⊢ 3 = (2 + 1)
4		7t2e14 12810	. 2 ⊢ (7 · 2) = ;14
5		1nn0 12512	. . 3 ⊢ 1 ∈ ℕ₀
6		4nn0 12515	. . 3 ⊢ 4 ∈ ℕ₀
7		eqid 2728	. . 3 ⊢ ;14 = ;14
8		1p1e2 12361	. . 3 ⊢ (1 + 1) = 2
9	1	nn0cni 12508	. . . 4 ⊢ 7 ∈ ℂ
10	6	nn0cni 12508	. . . 4 ⊢ 4 ∈ ℂ
11		7p4e11 12777	. . . 4 ⊢ (7 + 4) = ;11
12	9, 10, 11	addcomli 11430	. . 3 ⊢ (4 + 7) = ;11
13	5, 6, 1, 7, 8, 5, 12	decaddci 12762	. 2 ⊢ (;14 + 7) = ;21
14	1, 2, 3, 4, 13	4t3lem 12798	1 ⊢ (7 · 3) = ;21

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 (class class class)co 7414 1c1 11133 · cmul 11137 2c2 12291 3c3 12292 4c4 12293 7c7 12296 cdc 12701

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 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208

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 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 df-sub 11470 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-dec 12702

This theorem is referenced by: 7t4e28 12812 23prm 17081 prmlem2 17082 83prm 17085 163prm 17087 631prm 17089 1259prm 17098 log2ublem3 26873 log2ub 26874 ex-prmo 30262 hgt750lem2 34278 3exp7 41518 235t711 41861 ex-decpmul 41862 257prm 46895

	
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