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Theorem 4t3lem 8174
Description: Lemma for 4t3e12 8175 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
Hypotheses
Ref Expression
4t3lem.1 A 0
4t3lem.2 B 0
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 5466 . 2 (A · 𝐶) = (A · (B + 1))
3 4t3lem.1 . . . . . 6 A 0
43nn0cni 7929 . . . . 5 A
5 4t3lem.2 . . . . . 6 B 0
65nn0cni 7929 . . . . 5 B
7 ax-1cn 6736 . . . . 5 1
84, 6, 7adddii 6795 . . . 4 (A · (B + 1)) = ((A · B) + (A · 1))
9 4t3lem.4 . . . . 5 (A · B) = 𝐷
104mulid1i 6787 . . . . 5 (A · 1) = A
119, 10oveq12i 5467 . . . 4 ((A · B) + (A · 1)) = (𝐷 + A)
128, 11eqtri 2057 . . 3 (A · (B + 1)) = (𝐷 + A)
13 4t3lem.5 . . 3 (𝐷 + A) = 𝐸
1412, 13eqtri 2057 . 2 (A · (B + 1)) = 𝐸
152, 14eqtri 2057 1 (A · 𝐶) = 𝐸
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  (class class class)co 5455  1c1 6672   + caddc 6674   · cmul 6676  0cn0 7917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6734  ax-resscn 6735  ax-1cn 6736  ax-1re 6737  ax-icn 6738  ax-addcl 6739  ax-addrcl 6740  ax-mulcl 6741  ax-mulcom 6744  ax-mulass 6746  ax-distr 6747  ax-1rid 6750  ax-rnegex 6752  ax-cnre 6754
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458  df-inn 7656  df-n0 7918
This theorem is referenced by:  4t3e12  8175  4t4e16  8176  5t3e15  8177  5t4e20  8178  5t5e25  8179  6t3e18  8181  6t4e24  8182  6t5e30  8183  6t6e36  8184  7t3e21  8186  7t4e28  8187  7t5e35  8188  7t6e42  8189  7t7e49  8190  8t3e24  8192  8t4e32  8193  8t5e40  8194  8t6e48  8195  8t7e56  8196  8t8e64  8197  9t3e27  8199  9t4e36  8200  9t5e45  8201  9t6e54  8202  9t7e63  8203  9t8e72  8204  9t9e81  8205
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