6t5e30 - Intuitionistic Logic Explorer

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Theorem 6t5e30 9428

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 9135	. 2 ⊢ 6 ∈ ℕ₀
2		4nn0 9133	. 2 ⊢ 4 ∈ ℕ₀
3		df-5 8919	. 2 ⊢ 5 = (4 + 1)
4		6t4e24 9427	. 2 ⊢ (6 · 4) = ;24
5		2nn0 9131	. . 3 ⊢ 2 ∈ ℕ₀
6		eqid 2165	. . 3 ⊢ ;24 = ;24
7		2p1e3 8990	. . 3 ⊢ (2 + 1) = 3
8		6cn 8939	. . . 4 ⊢ 6 ∈ ℂ
9		4cn 8935	. . . 4 ⊢ 4 ∈ ℂ
10		6p4e10 9393	. . . 4 ⊢ (6 + 4) = ;10
11	8, 9, 10	addcomli 8043	. . 3 ⊢ (4 + 6) = ;10
12	5, 2, 1, 6, 7, 11	decaddci2 9383	. 2 ⊢ (;24 + 6) = ;30
13	1, 2, 3, 4, 12	4t3lem 9418	1 ⊢ (6 · 5) = ;30

Colors of variables: wff set class

Syntax hints: = wceq 1343 (class class class)co 5842 0cc0 7753 1c1 7754 · cmul 7758 2c2 8908 3c3 8909 4c4 8910 5c5 8911 6c6 8912 cdc 9322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-sub 8071 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-5 8919 df-6 8920 df-7 8921 df-8 8922 df-9 8923 df-n0 9115 df-dec 9323

This theorem is referenced by: 6t6e36 9429 5recm6rec 9465