6t5e30 - Intuitionistic Logic Explorer

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Theorem 6t5e30 9833

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 9534	. 2 ⊢ 6 ∈ ℕ₀
2		4nn0 9532	. 2 ⊢ 4 ∈ ℕ₀
3		df-5 9316	. 2 ⊢ 5 = (4 + 1)
4		6t4e24 9832	. 2 ⊢ (6 · 4) = ;24
5		2nn0 9530	. . 3 ⊢ 2 ∈ ℕ₀
6		eqid 2234	. . 3 ⊢ ;24 = ;24
7		2p1e3 9388	. . 3 ⊢ (2 + 1) = 3
8		6cn 9336	. . . 4 ⊢ 6 ∈ ℂ
9		4cn 9332	. . . 4 ⊢ 4 ∈ ℂ
10		6p4e10 9798	. . . 4 ⊢ (6 + 4) = ;10
11	8, 9, 10	addcomli 8434	. . 3 ⊢ (4 + 6) = ;10
12	5, 2, 1, 6, 7, 11	decaddci2 9788	. 2 ⊢ (;24 + 6) = ;30
13	1, 2, 3, 4, 12	4t3lem 9823	1 ⊢ (6 · 5) = ;30

Colors of variables: wff set class

Syntax hints: = wceq 1398 (class class class)co 6058 0cc0 8143 1c1 8144 · cmul 8148 2c2 9305 3c3 9306 4c4 9307 5c5 9308 6c6 9309 cdc 9727

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-br 4115 df-opab 4177 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-iota 5317 df-fun 5359 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-sub 8462 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-5 9316 df-6 9317 df-7 9318 df-8 9319 df-9 9320 df-n0 9514 df-dec 9728

This theorem is referenced by: 6t6e36 9834 5recm6rec 9870 2exp16 13160