6t5e30 - Intuitionistic Logic Explorer

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Theorem 6t5e30 9515

Description: 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)

**Assertion**
Ref	Expression
6t5e30	⊢ (6 · 5) = ;30

**Proof of Theorem 6t5e30**
Step	Hyp	Ref	Expression
1		6nn0 9222	. 2 ⊢ 6 ∈ ℕ₀
2		4nn0 9220	. 2 ⊢ 4 ∈ ℕ₀
3		df-5 9006	. 2 ⊢ 5 = (4 + 1)
4		6t4e24 9514	. 2 ⊢ (6 · 4) = ;24
5		2nn0 9218	. . 3 ⊢ 2 ∈ ℕ₀
6		eqid 2189	. . 3 ⊢ ;24 = ;24
7		2p1e3 9077	. . 3 ⊢ (2 + 1) = 3
8		6cn 9026	. . . 4 ⊢ 6 ∈ ℂ
9		4cn 9022	. . . 4 ⊢ 4 ∈ ℂ
10		6p4e10 9480	. . . 4 ⊢ (6 + 4) = ;10
11	8, 9, 10	addcomli 8127	. . 3 ⊢ (4 + 6) = ;10
12	5, 2, 1, 6, 7, 11	decaddci2 9470	. 2 ⊢ (;24 + 6) = ;30
13	1, 2, 3, 4, 12	4t3lem 9505	1 ⊢ (6 · 5) = ;30

Colors of variables: wff set class

Syntax hints: = wceq 1364 (class class class)co 5892 0cc0 7836 1c1 7837 · cmul 7841 2c2 8995 3c3 8996 4c4 8997 5c5 8998 6c6 8999 cdc 9409

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-cnre 7947

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-sub 8155 df-inn 8945 df-2 9003 df-3 9004 df-4 9005 df-5 9006 df-6 9007 df-7 9008 df-8 9009 df-9 9010 df-n0 9202 df-dec 9410

This theorem is referenced by: 6t6e36 9516 5recm6rec 9552