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Theorem 4t3lem 8724
Description: Lemma for 4t3e12 8725 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
StepHypRef Expression
1 4t3lem.3 . . 3 𝐶 = (𝐵 + 1)
21oveq2i 5575 . 2 (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1))
3 4t3lem.1 . . . . . 6 𝐴 ∈ ℕ0
43nn0cni 8437 . . . . 5 𝐴 ∈ ℂ
5 4t3lem.2 . . . . . 6 𝐵 ∈ ℕ0
65nn0cni 8437 . . . . 5 𝐵 ∈ ℂ
7 ax-1cn 7201 . . . . 5 1 ∈ ℂ
84, 6, 7adddii 7261 . . . 4 (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1))
9 4t3lem.4 . . . . 5 (𝐴 · 𝐵) = 𝐷
104mulid1i 7253 . . . . 5 (𝐴 · 1) = 𝐴
119, 10oveq12i 5576 . . . 4 ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴)
128, 11eqtri 2103 . . 3 (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴)
13 4t3lem.5 . . 3 (𝐷 + 𝐴) = 𝐸
1412, 13eqtri 2103 . 2 (𝐴 · (𝐵 + 1)) = 𝐸
152, 14eqtri 2103 1 (𝐴 · 𝐶) = 𝐸
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434  (class class class)co 5564  1c1 7114   + caddc 7116   · cmul 7118  0cn0 8425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulcom 7209  ax-mulass 7211  ax-distr 7212  ax-1rid 7215  ax-rnegex 7217  ax-cnre 7219
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5567  df-inn 8177  df-n0 8426
This theorem is referenced by:  4t3e12  8725  4t4e16  8726  5t2e10  8727  5t3e15  8728  5t4e20  8729  5t5e25  8730  6t3e18  8732  6t4e24  8733  6t5e30  8734  6t6e36  8735  7t3e21  8737  7t4e28  8738  7t5e35  8739  7t6e42  8740  7t7e49  8741  8t3e24  8743  8t4e32  8744  8t5e40  8745  8t6e48  8746  8t7e56  8747  8t8e64  8748  9t3e27  8750  9t4e36  8751  9t5e45  8752  9t6e54  8753  9t7e63  8754  9t8e72  8755  9t9e81  8756
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