8t5e40 - Intuitionistic Logic Explorer

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Theorem 8t5e40 9503

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 9201	. 2 ⊢ 8 ∈ ℕ₀
2		4nn0 9197	. 2 ⊢ 4 ∈ ℕ₀
3		df-5 8983	. 2 ⊢ 5 = (4 + 1)
4		8t4e32 9502	. 2 ⊢ (8 · 4) = ;32
5		3nn0 9196	. . 3 ⊢ 3 ∈ ℕ₀
6		2nn0 9195	. . 3 ⊢ 2 ∈ ℕ₀
7		eqid 2177	. . 3 ⊢ ;32 = ;32
8		3p1e4 9056	. . 3 ⊢ (3 + 1) = 4
9		8cn 9007	. . . 4 ⊢ 8 ∈ ℂ
10		2cn 8992	. . . 4 ⊢ 2 ∈ ℂ
11		8p2e10 9465	. . . 4 ⊢ (8 + 2) = ;10
12	9, 10, 11	addcomli 8104	. . 3 ⊢ (2 + 8) = ;10
13	5, 6, 1, 7, 8, 12	decaddci2 9447	. 2 ⊢ (;32 + 8) = ;40
14	1, 2, 3, 4, 13	4t3lem 9482	1 ⊢ (8 · 5) = ;40

Colors of variables: wff set class

Syntax hints: = wceq 1353 (class class class)co 5877 0cc0 7813 1c1 7814 · cmul 7818 2c2 8972 3c3 8973 4c4 8974 5c5 8975 8c8 8978 cdc 9386

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-sub 8132 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-7 8985 df-8 8986 df-9 8987 df-n0 9179 df-dec 9387

This theorem is referenced by: 8t6e48 9504