Theorem 4t3lem 9685

Description: Lemma for 4t3e12 9686 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 6018	. 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1))
3		4t3lem.1	. . . . . 6 ⊢ 𝐴 ∈ ℕ0
4	3	nn0cni 9392	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		4t3lem.2	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
6	5	nn0cni 9392	. . . . 5 ⊢ 𝐵 ∈ ℂ
7		ax-1cn 8103	. . . . 5 ⊢ 1 ∈ ℂ
8	4, 6, 7	adddii 8167	. . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1))
9		4t3lem.4	. . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷
10	4	mulridi 8159	. . . . 5 ⊢ (𝐴 · 1) = 𝐴
11	9, 10	oveq12i 6019	. . . 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 6007  1c1 8011   + caddc 8013   · cmul 8015  ℕ₀cn0 9380

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 4202  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulcom 8111  ax-mulass 8113  ax-distr 8114  ax-1rid 8117  ax-rnegex 8119  ax-cnre 8121

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 3889  df-int 3924  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6010  df-inn 9122  df-n0 9381

This theorem is referenced by:  4t3e12  9686  4t4e16  9687  5t2e10  9688  5t3e15  9689  5t4e20  9690  5t5e25  9691  6t3e18  9693  6t4e24  9694  6t5e30  9695  6t6e36  9696  7t3e21  9698  7t4e28  9699  7t5e35  9700  7t6e42  9701  7t7e49  9702  8t3e24  9704  8t4e32  9705  8t5e40  9706  8t6e48  9707  8t7e56  9708  8t8e64  9709  9t3e27  9711  9t4e36  9712  9t5e45  9713  9t6e54  9714  9t7e63  9715  9t8e72  9716  9t9e81  9717