6t5e30 - Intuitionistic Logic Explorer

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Theorem 6t5e30 9557

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 9264	. 2 ⊢ 6 ∈ ℕ₀
2		4nn0 9262	. 2 ⊢ 4 ∈ ℕ₀
3		df-5 9046	. 2 ⊢ 5 = (4 + 1)
4		6t4e24 9556	. 2 ⊢ (6 · 4) = ;24
5		2nn0 9260	. . 3 ⊢ 2 ∈ ℕ₀
6		eqid 2193	. . 3 ⊢ ;24 = ;24
7		2p1e3 9118	. . 3 ⊢ (2 + 1) = 3
8		6cn 9066	. . . 4 ⊢ 6 ∈ ℂ
9		4cn 9062	. . . 4 ⊢ 4 ∈ ℂ
10		6p4e10 9522	. . . 4 ⊢ (6 + 4) = ;10
11	8, 9, 10	addcomli 8166	. . 3 ⊢ (4 + 6) = ;10
12	5, 2, 1, 6, 7, 11	decaddci2 9512	. 2 ⊢ (;24 + 6) = ;30
13	1, 2, 3, 4, 12	4t3lem 9547	1 ⊢ (6 · 5) = ;30

Colors of variables: wff set class

Syntax hints: = wceq 1364 (class class class)co 5919 0cc0 7874 1c1 7875 · cmul 7879 2c2 9035 3c3 9036 4c4 9037 5c5 9038 6c6 9039 cdc 9451

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-br 4031 df-opab 4092 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-sub 8194 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-9 9050 df-n0 9244 df-dec 9452

This theorem is referenced by: 6t6e36 9558 5recm6rec 9594