Theorem 4t3lem 9697

Description: Lemma for 4t3e12 9698 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 6024	. 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1))
3		4t3lem.1	. . . . . 6 ⊢ 𝐴 ∈ ℕ0
4	3	nn0cni 9404	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		4t3lem.2	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
6	5	nn0cni 9404	. . . . 5 ⊢ 𝐵 ∈ ℂ
7		ax-1cn 8115	. . . . 5 ⊢ 1 ∈ ℂ
8	4, 6, 7	adddii 8179	. . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1))
9		4t3lem.4	. . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷
10	4	mulridi 8171	. . . . 5 ⊢ (𝐴 · 1) = 𝐴
11	9, 10	oveq12i 6025	. . . 4 ⊢ ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴)
12	8, 11	eqtri 2250	. . 3 ⊢ (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴)
13		4t3lem.5	. . 3 ⊢ (𝐷 + 𝐴) = 𝐸
14	12, 13	eqtri 2250	. 2 ⊢ (𝐴 · (𝐵 + 1)) = 𝐸
15	2, 14	eqtri 2250	1 ⊢ (𝐴 · 𝐶) = 𝐸

Syntax hints:   = wceq 1395   ∈ wcel 2200  (class class class)co 6013  1c1 8023   + caddc 8025   · cmul 8027  ℕ₀cn0 9392

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4205  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulcom 8123  ax-mulass 8125  ax-distr 8126  ax-1rid 8129  ax-rnegex 8131  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-iota 5284  df-fv 5332  df-ov 6016  df-inn 9134  df-n0 9393

This theorem is referenced by:  4t3e12  9698  4t4e16  9699  5t2e10  9700  5t3e15  9701  5t4e20  9702  5t5e25  9703  6t3e18  9705  6t4e24  9706  6t5e30  9707  6t6e36  9708  7t3e21  9710  7t4e28  9711  7t5e35  9712  7t6e42  9713  7t7e49  9714  8t3e24  9716  8t4e32  9717  8t5e40  9718  8t6e48  9719  8t7e56  9720  8t8e64  9721  9t3e27  9723  9t4e36  9724  9t5e45  9725  9t6e54  9726  9t7e63  9727  9t8e72  9728  9t9e81  9729