Theorem 4t3lem 9600

Description: Lemma for 4t3e12 9601 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)

Hypotheses

Ref	Expression
4t3lem.1	|- A e. NN0
4t3lem.2	|- B e. NN0
4t3lem.3	|- C = ( B + 1 )
4t3lem.4	|- ( A x. B ) = D
4t3lem.5	|- ( D + A ) = E

Assertion

Ref	Expression
4t3lem	|- ( A x. C ) = E

Proof of Theorem 4t3lem

Step	Hyp	Ref	Expression
1		4t3lem.3	. . 3 |- C = ( B + 1 )
2	1	oveq2i 5955	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9307	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9307	. . . . 5 |- B e. CC
7		ax-1cn 8018	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8082	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8074	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5956	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2226	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2226	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2226	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1373   e. wcel 2176  (class class class)co 5944   1c1 7926   + caddc 7928   x. cmul 7930   NN0cn0 9295

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-sep 4162  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulcom 8026  ax-mulass 8028  ax-distr 8029  ax-1rid 8032  ax-rnegex 8034  ax-cnre 8036

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947  df-inn 9037  df-n0 9296

This theorem is referenced by:  4t3e12  9601  4t4e16  9602  5t2e10  9603  5t3e15  9604  5t4e20  9605  5t5e25  9606  6t3e18  9608  6t4e24  9609  6t5e30  9610  6t6e36  9611  7t3e21  9613  7t4e28  9614  7t5e35  9615  7t6e42  9616  7t7e49  9617  8t3e24  9619  8t4e32  9620  8t5e40  9621  8t6e48  9622  8t7e56  9623  8t8e64  9624  9t3e27  9626  9t4e36  9627  9t5e45  9628  9t6e54  9629  9t7e63  9630  9t8e72  9631  9t9e81  9632