8t5e40 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 8t5e40		GIF version

Theorem 8t5e40 9496

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 9194	. 2 ⊢ 8 ∈ ℕ₀
2		4nn0 9190	. 2 ⊢ 4 ∈ ℕ₀
3		df-5 8976	. 2 ⊢ 5 = (4 + 1)
4		8t4e32 9495	. 2 ⊢ (8 · 4) = ;32
5		3nn0 9189	. . 3 ⊢ 3 ∈ ℕ₀
6		2nn0 9188	. . 3 ⊢ 2 ∈ ℕ₀
7		eqid 2177	. . 3 ⊢ ;32 = ;32
8		3p1e4 9049	. . 3 ⊢ (3 + 1) = 4
9		8cn 9000	. . . 4 ⊢ 8 ∈ ℂ
10		2cn 8985	. . . 4 ⊢ 2 ∈ ℂ
11		8p2e10 9458	. . . 4 ⊢ (8 + 2) = ;10
12	9, 10, 11	addcomli 8097	. . 3 ⊢ (2 + 8) = ;10
13	5, 6, 1, 7, 8, 12	decaddci2 9440	. 2 ⊢ (;32 + 8) = ;40
14	1, 2, 3, 4, 13	4t3lem 9475	1 ⊢ (8 · 5) = ;40

Colors of variables: wff set class

Syntax hints: = wceq 1353 (class class class)co 5871 0cc0 7807 1c1 7808 · cmul 7812 2c2 8965 3c3 8966 4c4 8967 5c5 8968 8c8 8971 cdc 9379

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 4120 ax-pow 4173 ax-pr 4208 ax-setind 4535 ax-cnex 7898 ax-resscn 7899 ax-1cn 7900 ax-1re 7901 ax-icn 7902 ax-addcl 7903 ax-addrcl 7904 ax-mulcl 7905 ax-addcom 7907 ax-mulcom 7908 ax-addass 7909 ax-mulass 7910 ax-distr 7911 ax-i2m1 7912 ax-1rid 7914 ax-0id 7915 ax-rnegex 7916 ax-cnre 7918

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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5176 df-fun 5216 df-fv 5222 df-riota 5827 df-ov 5874 df-oprab 5875 df-mpo 5876 df-sub 8125 df-inn 8915 df-2 8973 df-3 8974 df-4 8975 df-5 8976 df-6 8977 df-7 8978 df-8 8979 df-9 8980 df-n0 9172 df-dec 9380

This theorem is referenced by: 8t6e48 9497