Theorem 4t3lem 9823

Description: Lemma for 4t3e12 9824 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 6069	. 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1))
3		4t3lem.1	. . . . . 6 ⊢ 𝐴 ∈ ℕ0
4	3	nn0cni 9525	. . . . 5 ⊢ 𝐴 ∈ ℂ
5		4t3lem.2	. . . . . 6 ⊢ 𝐵 ∈ ℕ0
6	5	nn0cni 9525	. . . . 5 ⊢ 𝐵 ∈ ℂ
7		ax-1cn 8236	. . . . 5 ⊢ 1 ∈ ℂ
8	4, 6, 7	adddii 8300	. . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1))
9		4t3lem.4	. . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷
10	4	mulridi 8292	. . . . 5 ⊢ (𝐴 · 1) = 𝐴
11	9, 10	oveq12i 6070	. . . 4 ⊢ ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴)
12	8, 11	eqtri 2255	. . 3 ⊢ (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴)
13		4t3lem.5	. . 3 ⊢ (𝐷 + 𝐴) = 𝐸
14	12, 13	eqtri 2255	. 2 ⊢ (𝐴 · (𝐵 + 1)) = 𝐸
15	2, 14	eqtri 2255	1 ⊢ (𝐴 · 𝐶) = 𝐸

Syntax hints:   = wceq 1398   ∈ wcel 2205  (class class class)co 6058  1c1 8144   + caddc 8146   · cmul 8148  ℕ₀cn0 9513

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-sep 4233  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulcom 8244  ax-mulass 8246  ax-distr 8247  ax-1rid 8250  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-inn 9255  df-n0 9514

This theorem is referenced by:  4t3e12  9824  4t4e16  9825  5t2e10  9826  5t3e15  9827  5t4e20  9828  5t5e25  9829  6t3e18  9831  6t4e24  9832  6t5e30  9833  6t6e36  9834  7t3e21  9836  7t4e28  9837  7t5e35  9838  7t6e42  9839  7t7e49  9840  8t3e24  9842  8t4e32  9843  8t5e40  9844  8t6e48  9845  8t7e56  9846  8t8e64  9847  9t3e27  9849  9t4e36  9850  9t5e45  9851  9t6e54  9852  9t7e63  9853  9t8e72  9854  9t9e81  9855