Theorem 4t3lem 9706

Description: Lemma for 4t3e12 9707 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 6028	. 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1))
3		4t3lem.1	. . . . . 6 ⊢ 𝐴 ∈ ℕ0
4	3	nn0cni 9413	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		4t3lem.2	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
6	5	nn0cni 9413	. . . . 5 ⊢ 𝐵 ∈ ℂ
7		ax-1cn 8124	. . . . 5 ⊢ 1 ∈ ℂ
8	4, 6, 7	adddii 8188	. . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1))
9		4t3lem.4	. . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷
10	4	mulridi 8180	. . . . 5 ⊢ (𝐴 · 1) = 𝐴
11	9, 10	oveq12i 6029	. . . 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 6017  1c1 8032   + caddc 8034   · cmul 8036  ℕ₀cn0 9401

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulcom 8132  ax-mulass 8134  ax-distr 8135  ax-1rid 8138  ax-rnegex 8140  ax-cnre 8142

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 6020  df-inn 9143  df-n0 9402

This theorem is referenced by:  4t3e12  9707  4t4e16  9708  5t2e10  9709  5t3e15  9710  5t4e20  9711  5t5e25  9712  6t3e18  9714  6t4e24  9715  6t5e30  9716  6t6e36  9717  7t3e21  9719  7t4e28  9720  7t5e35  9721  7t6e42  9722  7t7e49  9723  8t3e24  9725  8t4e32  9726  8t5e40  9727  8t6e48  9728  8t7e56  9729  8t8e64  9730  9t3e27  9732  9t4e36  9733  9t5e45  9734  9t6e54  9735  9t7e63  9736  9t8e72  9737  9t9e81  9738