8t5e40 - Intuitionistic Logic Explorer

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Theorem 8t5e40 9460

Description: 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)

**Assertion**
Ref	Expression
8t5e40	⊢ (8 · 5) = ;40

**Proof of Theorem 8t5e40**
Step	Hyp	Ref	Expression
1		8nn0 9158	. 2 ⊢ 8 ∈ ℕ₀
2		4nn0 9154	. 2 ⊢ 4 ∈ ℕ₀
3		df-5 8940	. 2 ⊢ 5 = (4 + 1)
4		8t4e32 9459	. 2 ⊢ (8 · 4) = ;32
5		3nn0 9153	. . 3 ⊢ 3 ∈ ℕ₀
6		2nn0 9152	. . 3 ⊢ 2 ∈ ℕ₀
7		eqid 2170	. . 3 ⊢ ;32 = ;32
8		3p1e4 9013	. . 3 ⊢ (3 + 1) = 4
9		8cn 8964	. . . 4 ⊢ 8 ∈ ℂ
10		2cn 8949	. . . 4 ⊢ 2 ∈ ℂ
11		8p2e10 9422	. . . 4 ⊢ (8 + 2) = ;10
12	9, 10, 11	addcomli 8064	. . 3 ⊢ (2 + 8) = ;10
13	5, 6, 1, 7, 8, 12	decaddci2 9404	. 2 ⊢ (;32 + 8) = ;40
14	1, 2, 3, 4, 13	4t3lem 9439	1 ⊢ (8 · 5) = ;40

Colors of variables: wff set class

Syntax hints: = wceq 1348 (class class class)co 5853 0cc0 7774 1c1 7775 · cmul 7779 2c2 8929 3c3 8930 4c4 8931 5c5 8932 8c8 8935 cdc 9343

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-9 8944 df-n0 9136 df-dec 9344

This theorem is referenced by: 8t6e48 9461