Theorem 4t3lem 9751

Description: Lemma for 4t3e12 9752 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)

Hypotheses

Ref	Expression
4t3lem.1	⊢ 𝐴 ∈ ℕ0
4t3lem.2	⊢ 𝐵 ∈ ℕ0
4t3lem.3	⊢ 𝐶 = (𝐵 + 1)
4t3lem.4	⊢ (𝐴 · 𝐵) = 𝐷
4t3lem.5	⊢ (𝐷 + 𝐴) = 𝐸

Assertion

Ref	Expression
4t3lem	⊢ (𝐴 · 𝐶) = 𝐸

Proof of Theorem 4t3lem

Step	Hyp	Ref	Expression
1		4t3lem.3	. . 3 ⊢ 𝐶 = (𝐵 + 1)
2	1	oveq2i 6039	. 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1))
3		4t3lem.1	. . . . . 6 ⊢ 𝐴 ∈ ℕ0
4	3	nn0cni 9456	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		4t3lem.2	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
6	5	nn0cni 9456	. . . . 5 ⊢ 𝐵 ∈ ℂ
7		ax-1cn 8168	. . . . 5 ⊢ 1 ∈ ℂ
8	4, 6, 7	adddii 8232	. . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1))
9		4t3lem.4	. . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷
10	4	mulridi 8224	. . . . 5 ⊢ (𝐴 · 1) = 𝐴
11	9, 10	oveq12i 6040	. . . 4 ⊢ ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴)
12	8, 11	eqtri 2252	. . 3 ⊢ (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴)
13		4t3lem.5	. . 3 ⊢ (𝐷 + 𝐴) = 𝐸
14	12, 13	eqtri 2252	. 2 ⊢ (𝐴 · (𝐵 + 1)) = 𝐸
15	2, 14	eqtri 2252	1 ⊢ (𝐴 · 𝐶) = 𝐸

Syntax hints:   = wceq 1398   ∈ wcel 2202  (class class class)co 6028  1c1 8076   + caddc 8078   · cmul 8080  ℕ₀cn0 9444

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-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-ext 2213  ax-sep 4212  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulcom 8176  ax-mulass 8178  ax-distr 8179  ax-1rid 8182  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031  df-inn 9186  df-n0 9445

This theorem is referenced by:  4t3e12  9752  4t4e16  9753  5t2e10  9754  5t3e15  9755  5t4e20  9756  5t5e25  9757  6t3e18  9759  6t4e24  9760  6t5e30  9761  6t6e36  9762  7t3e21  9764  7t4e28  9765  7t5e35  9766  7t6e42  9767  7t7e49  9768  8t3e24  9770  8t4e32  9771  8t5e40  9772  8t6e48  9773  8t7e56  9774  8t8e64  9775  9t3e27  9777  9t4e36  9778  9t5e45  9779  9t6e54  9780  9t7e63  9781  9t8e72  9782  9t9e81  9783