6t5e30 - Intuitionistic Logic Explorer

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Theorem 6t5e30 9419

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 9126	. 2 ⊢ 6 ∈ ℕ₀
2		4nn0 9124	. 2 ⊢ 4 ∈ ℕ₀
3		df-5 8910	. 2 ⊢ 5 = (4 + 1)
4		6t4e24 9418	. 2 ⊢ (6 · 4) = ;24
5		2nn0 9122	. . 3 ⊢ 2 ∈ ℕ₀
6		eqid 2164	. . 3 ⊢ ;24 = ;24
7		2p1e3 8981	. . 3 ⊢ (2 + 1) = 3
8		6cn 8930	. . . 4 ⊢ 6 ∈ ℂ
9		4cn 8926	. . . 4 ⊢ 4 ∈ ℂ
10		6p4e10 9384	. . . 4 ⊢ (6 + 4) = ;10
11	8, 9, 10	addcomli 8034	. . 3 ⊢ (4 + 6) = ;10
12	5, 2, 1, 6, 7, 11	decaddci2 9374	. 2 ⊢ (;24 + 6) = ;30
13	1, 2, 3, 4, 12	4t3lem 9409	1 ⊢ (6 · 5) = ;30

Colors of variables: wff set class

Syntax hints: = wceq 1342 (class class class)co 5836 0cc0 7744 1c1 7745 · cmul 7749 2c2 8899 3c3 8900 4c4 8901 5c5 8902 6c6 8903 cdc 9313

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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855

This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-dec 9314

This theorem is referenced by: 6t6e36 9420 5recm6rec 9456