Theorem 4t3lem 9635

Description: Lemma for 4t3e12 9636 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 5978	. 2 |- ( A x. C ) = ( A x. ( B + 1 ) )
3		4t3lem.1	. . . . . 6 |- A e. NN0
4	3	nn0cni 9342	. . . . 5 |- A e. CC
5		4t3lem.2	. . . . . 6 |- B e. NN0
6	5	nn0cni 9342	. . . . 5 |- B e. CC
7		ax-1cn 8053	. . . . 5 |- 1 e. CC
8	4, 6, 7	adddii 8117	. . . 4 |- ( A x. ( B + 1 ) ) = ( ( A x. B ) + ( A x. 1 ) )
9		4t3lem.4	. . . . 5 |- ( A x. B ) = D
10	4	mulridi 8109	. . . . 5 |- ( A x. 1 ) = A
11	9, 10	oveq12i 5979	. . . 4 |- ( ( A x. B ) + ( A x. 1 ) ) = ( D + A )
12	8, 11	eqtri 2228	. . 3 |- ( A x. ( B + 1 ) ) = ( D + A )
13		4t3lem.5	. . 3 |- ( D + A ) = E
14	12, 13	eqtri 2228	. 2 |- ( A x. ( B + 1 ) ) = E
15	2, 14	eqtri 2228	1 |- ( A x. C ) = E

Syntax hints:   = wceq 1373   e. wcel 2178  (class class class)co 5967   1c1 7961   + caddc 7963   x. cmul 7965   NN0cn0 9330

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-sep 4178  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulcom 8061  ax-mulass 8063  ax-distr 8064  ax-1rid 8067  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970  df-inn 9072  df-n0 9331

This theorem is referenced by:  4t3e12  9636  4t4e16  9637  5t2e10  9638  5t3e15  9639  5t4e20  9640  5t5e25  9641  6t3e18  9643  6t4e24  9644  6t5e30  9645  6t6e36  9646  7t3e21  9648  7t4e28  9649  7t5e35  9650  7t6e42  9651  7t7e49  9652  8t3e24  9654  8t4e32  9655  8t5e40  9656  8t6e48  9657  8t7e56  9658  8t8e64  9659  9t3e27  9661  9t4e36  9662  9t5e45  9663  9t6e54  9664  9t7e63  9665  9t8e72  9666  9t9e81  9667