Theorem 4t3lem 9707

Description: Lemma for 4t3e12 9708 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 6029	. 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1))
3		4t3lem.1	. . . . . 6 ⊢ 𝐴 ∈ ℕ0
4	3	nn0cni 9414	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		4t3lem.2	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
6	5	nn0cni 9414	. . . . 5 ⊢ 𝐵 ∈ ℂ
7		ax-1cn 8125	. . . . 5 ⊢ 1 ∈ ℂ
8	4, 6, 7	adddii 8189	. . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1))
9		4t3lem.4	. . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷
10	4	mulridi 8181	. . . . 5 ⊢ (𝐴 · 1) = 𝐴
11	9, 10	oveq12i 6030	. . . 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 1397   ∈ wcel 2202  (class class class)co 6018  1c1 8033   + caddc 8035   · cmul 8037  ℕ₀cn0 9402

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulcom 8133  ax-mulass 8135  ax-distr 8136  ax-1rid 8139  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021  df-inn 9144  df-n0 9403

This theorem is referenced by:  4t3e12  9708  4t4e16  9709  5t2e10  9710  5t3e15  9711  5t4e20  9712  5t5e25  9713  6t3e18  9715  6t4e24  9716  6t5e30  9717  6t6e36  9718  7t3e21  9720  7t4e28  9721  7t5e35  9722  7t6e42  9723  7t7e49  9724  8t3e24  9726  8t4e32  9727  8t5e40  9728  8t6e48  9729  8t7e56  9730  8t8e64  9731  9t3e27  9733  9t4e36  9734  9t5e45  9735  9t6e54  9736  9t7e63  9737  9t8e72  9738  9t9e81  9739