Theorem 4t3lem 9670

Description: Lemma for 4t3e12 9671 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 6011	. 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1))
3		4t3lem.1	. . . . . 6 ⊢ 𝐴 ∈ ℕ0
4	3	nn0cni 9377	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		4t3lem.2	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
6	5	nn0cni 9377	. . . . 5 ⊢ 𝐵 ∈ ℂ
7		ax-1cn 8088	. . . . 5 ⊢ 1 ∈ ℂ
8	4, 6, 7	adddii 8152	. . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1))
9		4t3lem.4	. . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷
10	4	mulridi 8144	. . . . 5 ⊢ (𝐴 · 1) = 𝐴
11	9, 10	oveq12i 6012	. . . 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 6000  1c1 7996   + caddc 7998   · cmul 8000  ℕ₀cn0 9365

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 4201  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulcom 8096  ax-mulass 8098  ax-distr 8099  ax-1rid 8102  ax-rnegex 8104  ax-cnre 8106

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003  df-inn 9107  df-n0 9366

This theorem is referenced by:  4t3e12  9671  4t4e16  9672  5t2e10  9673  5t3e15  9674  5t4e20  9675  5t5e25  9676  6t3e18  9678  6t4e24  9679  6t5e30  9680  6t6e36  9681  7t3e21  9683  7t4e28  9684  7t5e35  9685  7t6e42  9686  7t7e49  9687  8t3e24  9689  8t4e32  9690  8t5e40  9691  8t6e48  9692  8t7e56  9693  8t8e64  9694  9t3e27  9696  9t4e36  9697  9t5e45  9698  9t6e54  9699  9t7e63  9700  9t8e72  9701  9t9e81  9702