Theorem problem4 31562

Description: Practice problem 4. Clues: pm3.2i 471 eqcomi 2631 eqtri 2644 subaddrii 10370 recni 10052 7re 11103 6re 11101 ax-1cn 9994 df-7 11084 ax-mp 5 oveq1i 6660 3cn 11095 2cn 11091 df-3 11080 mulid2i 10043 subdiri 10480 mp3an 1424 mulcli 10045 subadd23 10293 oveq2i 6661 oveq12i 6662 3t2e6 11179 mulcomi 10046 subcli 10357 biimpri 218 subadd2i 10369. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
problem4.1	|- A e. CC
problem4.2	|- B e. CC
problem4.3	|- ( A + B ) = 3
problem4.4	|- ( ( 3 x. A ) + ( 2 x. B ) ) = 7

Assertion

Ref	Expression
problem4	|- ( A = 1 /\ B = 2 )

Proof of Theorem problem4

Step	Hyp	Ref	Expression
1		7re 11103	. . . . . . 7 |- 7 e. RR
2	1	recni 10052	. . . . . 6 |- 7 e. CC
3		6re 11101	. . . . . . 7 |- 6 e. RR
4	3	recni 10052	. . . . . 6 |- 6 e. CC
5		ax-1cn 9994	. . . . . 6 |- 1 e. CC
6		df-7 11084	. . . . . . 7 |- 7 = ( 6 + 1 )
7	6	eqcomi 2631	. . . . . 6 |- ( 6 + 1 ) = 7
8	2, 4, 5, 7	subaddrii 10370	. . . . 5 |- ( 7 - 6 ) = 1
9	8	eqcomi 2631	. . . 4 |- 1 = ( 7 - 6 )
10		3cn 11095	. . . . . . . . . . . . 13 |- 3 e. CC
11		2cn 11091	. . . . . . . . . . . . 13 |- 2 e. CC
12		df-3 11080	. . . . . . . . . . . . . 14 |- 3 = ( 2 + 1 )
13	12	eqcomi 2631	. . . . . . . . . . . . 13 |- ( 2 + 1 ) = 3
14	10, 11, 5, 13	subaddrii 10370	. . . . . . . . . . . 12 |- ( 3 - 2 ) = 1
15	14	oveq1i 6660	. . . . . . . . . . 11 |- ( ( 3 - 2 ) x. A ) = ( 1 x. A )
16		problem4.1	. . . . . . . . . . . 12 |- A e. CC
17	16	mulid2i 10043	. . . . . . . . . . 11 |- ( 1 x. A ) = A
18	15, 17	eqtri 2644	. . . . . . . . . 10 |- ( ( 3 - 2 ) x. A ) = A
19	18	eqcomi 2631	. . . . . . . . 9 |- A = ( ( 3 - 2 ) x. A )
20	10, 11, 16	subdiri 10480	. . . . . . . . 9 |- ( ( 3 - 2 ) x. A ) = ( ( 3 x. A ) - ( 2 x. A ) )
21	19, 20	eqtri 2644	. . . . . . . 8 |- A = ( ( 3 x. A ) - ( 2 x. A ) )
22	21	oveq1i 6660	. . . . . . 7 |- ( A + 6 ) = ( ( ( 3 x. A ) - ( 2 x. A ) ) + 6 )
23	10, 16	mulcli 10045	. . . . . . . . 9 |- ( 3 x. A ) e. CC
24	11, 16	mulcli 10045	. . . . . . . . 9 |- ( 2 x. A ) e. CC
25		subadd23 10293	. . . . . . . . 9 |- ( ( ( 3 x. A ) e. CC /\ ( 2 x. A ) e. CC /\ 6 e. CC ) -> ( ( ( 3 x. A ) - ( 2 x. A ) ) + 6 ) = ( ( 3 x. A ) + ( 6 - ( 2 x. A ) ) ) )
26	23, 24, 4, 25	mp3an 1424	. . . . . . . 8 |- ( ( ( 3 x. A ) - ( 2 x. A ) ) + 6 ) = ( ( 3 x. A ) + ( 6 - ( 2 x. A ) ) )
27		3t2e6 11179	. . . . . . . . . . . . . 14 |- ( 3 x. 2 ) = 6
28	16, 11	mulcomi 10046	. . . . . . . . . . . . . 14 |- ( A x. 2 ) = ( 2 x. A )
29	27, 28	oveq12i 6662	. . . . . . . . . . . . 13 |- ( ( 3 x. 2 ) - ( A x. 2 ) ) = ( 6 - ( 2 x. A ) )
30	29	eqcomi 2631	. . . . . . . . . . . 12 |- ( 6 - ( 2 x. A ) ) = ( ( 3 x. 2 ) - ( A x. 2 ) )
31	10, 16, 11	subdiri 10480	. . . . . . . . . . . . . 14 |- ( ( 3 - A ) x. 2 ) = ( ( 3 x. 2 ) - ( A x. 2 ) )
32	31	eqcomi 2631	. . . . . . . . . . . . 13 |- ( ( 3 x. 2 ) - ( A x. 2 ) ) = ( ( 3 - A ) x. 2 )
33	10, 16	subcli 10357	. . . . . . . . . . . . . . . 16 |- ( 3 - A ) e. CC
34	11, 33	mulcomi 10046	. . . . . . . . . . . . . . 15 |- ( 2 x. ( 3 - A ) ) = ( ( 3 - A ) x. 2 )
35	34	eqcomi 2631	. . . . . . . . . . . . . 14 |- ( ( 3 - A ) x. 2 ) = ( 2 x. ( 3 - A ) )
36		problem4.2	. . . . . . . . . . . . . . . . . 18 |- B e. CC
37		problem4.3	. . . . . . . . . . . . . . . . . 18 |- ( A + B ) = 3
38	10, 16, 36, 37	subaddrii 10370	. . . . . . . . . . . . . . . . 17 |- ( 3 - A ) = B
39	38	eqcomi 2631	. . . . . . . . . . . . . . . 16 |- B = ( 3 - A )
40	39	oveq2i 6661	. . . . . . . . . . . . . . 15 |- ( 2 x. B ) = ( 2 x. ( 3 - A ) )
41	40	eqcomi 2631	. . . . . . . . . . . . . 14 |- ( 2 x. ( 3 - A ) ) = ( 2 x. B )
42	35, 41	eqtri 2644	. . . . . . . . . . . . 13 |- ( ( 3 - A ) x. 2 ) = ( 2 x. B )
43	32, 42	eqtri 2644	. . . . . . . . . . . 12 |- ( ( 3 x. 2 ) - ( A x. 2 ) ) = ( 2 x. B )
44	30, 43	eqtri 2644	. . . . . . . . . . 11 |- ( 6 - ( 2 x. A ) ) = ( 2 x. B )
45	44	eqcomi 2631	. . . . . . . . . 10 |- ( 2 x. B ) = ( 6 - ( 2 x. A ) )
46	45	oveq2i 6661	. . . . . . . . 9 |- ( ( 3 x. A ) + ( 2 x. B ) ) = ( ( 3 x. A ) + ( 6 - ( 2 x. A ) ) )
47	46	eqcomi 2631	. . . . . . . 8 |- ( ( 3 x. A ) + ( 6 - ( 2 x. A ) ) ) = ( ( 3 x. A ) + ( 2 x. B ) )
48	26, 47	eqtri 2644	. . . . . . 7 |- ( ( ( 3 x. A ) - ( 2 x. A ) ) + 6 ) = ( ( 3 x. A ) + ( 2 x. B ) )
49	22, 48	eqtri 2644	. . . . . 6 |- ( A + 6 ) = ( ( 3 x. A ) + ( 2 x. B ) )
50		problem4.4	. . . . . 6 |- ( ( 3 x. A ) + ( 2 x. B ) ) = 7
51	49, 50	eqtri 2644	. . . . 5 |- ( A + 6 ) = 7
52	2, 4, 16	subadd2i 10369	. . . . . 6 |- ( ( 7 - 6 ) = A <-> ( A + 6 ) = 7 )
53	52	biimpri 218	. . . . 5 |- ( ( A + 6 ) = 7 -> ( 7 - 6 ) = A )
54	51, 53	ax-mp 5	. . . 4 |- ( 7 - 6 ) = A
55	9, 54	eqtri 2644	. . 3 |- 1 = A
56	55	eqcomi 2631	. 2 |- A = 1
57	56	oveq2i 6661	. . . 4 |- ( 3 - A ) = ( 3 - 1 )
58	10, 5, 11	subadd2i 10369	. . . . . 6 |- ( ( 3 - 1 ) = 2 <-> ( 2 + 1 ) = 3 )
59	58	biimpri 218	. . . . 5 |- ( ( 2 + 1 ) = 3 -> ( 3 - 1 ) = 2 )
60	13, 59	ax-mp 5	. . . 4 |- ( 3 - 1 ) = 2
61	57, 60	eqtri 2644	. . 3 |- ( 3 - A ) = 2
62	39, 61	eqtri 2644	. 2 |- B = 2
63	56, 62	pm3.2i 471	1 |- ( A = 1 /\ B = 2 )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   2c2 11070   3c3 11071   6c6 11074   7c7 11075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084

This theorem is referenced by: (None)