Theorem 4t3lem 9418

Description: Lemma for 4t3e12 9419 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 5853	. 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1))
3		4t3lem.1	. . . . . 6 ⊢ 𝐴 ∈ ℕ0
4	3	nn0cni 9126	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		4t3lem.2	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
6	5	nn0cni 9126	. . . . 5 ⊢ 𝐵 ∈ ℂ
7		ax-1cn 7846	. . . . 5 ⊢ 1 ∈ ℂ
8	4, 6, 7	adddii 7909	. . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1))
9		4t3lem.4	. . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷
10	4	mulid1i 7901	. . . . 5 ⊢ (𝐴 · 1) = 𝐴
11	9, 10	oveq12i 5854	. . . 4 ⊢ ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴)
12	8, 11	eqtri 2186	. . 3 ⊢ (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴)
13		4t3lem.5	. . 3 ⊢ (𝐷 + 𝐴) = 𝐸
14	12, 13	eqtri 2186	. 2 ⊢ (𝐴 · (𝐵 + 1)) = 𝐸
15	2, 14	eqtri 2186	1 ⊢ (𝐴 · 𝐶) = 𝐸

Syntax hints:   = wceq 1343   ∈ wcel 2136  (class class class)co 5842  1c1 7754   + caddc 7756   · cmul 7758  ℕ₀cn0 9114

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulcom 7854  ax-mulass 7856  ax-distr 7857  ax-1rid 7860  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845  df-inn 8858  df-n0 9115

This theorem is referenced by:  4t3e12  9419  4t4e16  9420  5t2e10  9421  5t3e15  9422  5t4e20  9423  5t5e25  9424  6t3e18  9426  6t4e24  9427  6t5e30  9428  6t6e36  9429  7t3e21  9431  7t4e28  9432  7t5e35  9433  7t6e42  9434  7t7e49  9435  8t3e24  9437  8t4e32  9438  8t5e40  9439  8t6e48  9440  8t7e56  9441  8t8e64  9442  9t3e27  9444  9t4e36  9445  9t5e45  9446  9t6e54  9447  9t7e63  9448  9t8e72  9449  9t9e81  9450