Theorem sqeqd 13906

Description: A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)

Hypotheses

Ref	Expression
sqeqd.1	|- ( ph -> A e. CC )
sqeqd.2	|- ( ph -> B e. CC )
sqeqd.3	|- ( ph -> ( A ^ 2 ) = ( B ^ 2 ) )
sqeqd.4	|- ( ph -> 0 <_ ( Re ` A ) )
sqeqd.5	|- ( ph -> 0 <_ ( Re ` B ) )
sqeqd.6	|- ( ( ph /\ ( Re ` A ) = 0 /\ ( Re ` B ) = 0 ) -> A = B )

Assertion

Ref	Expression
sqeqd	|- ( ph -> A = B )

Proof of Theorem sqeqd

Step	Hyp	Ref	Expression
1		sqeqd.3	. . . . 5 |- ( ph -> ( A ^ 2 ) = ( B ^ 2 ) )
2		sqeqd.1	. . . . . 6 |- ( ph -> A e. CC )
3		sqeqd.2	. . . . . 6 |- ( ph -> B e. CC )
4		sqeqor 12978	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( A = B \/ A = -u B ) ) )
5	2, 3, 4	syl2anc 693	. . . . 5 |- ( ph -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( A = B \/ A = -u B ) ) )
6	1, 5	mpbid 222	. . . 4 |- ( ph -> ( A = B \/ A = -u B ) )
7	6	ord 392	. . 3 |- ( ph -> ( -. A = B -> A = -u B ) )
8		simpl 473	. . . . 5 |- ( ( ph /\ A = -u B ) -> ph )
9		fveq2 6191	. . . . . . 7 |- ( A = -u B -> ( Re ` A ) = ( Re ` -u B ) )
10		reneg 13865	. . . . . . . 8 |- ( B e. CC -> ( Re ` -u B ) = -u ( Re ` B ) )
11	3, 10	syl 17	. . . . . . 7 |- ( ph -> ( Re ` -u B ) = -u ( Re ` B ) )
12	9, 11	sylan9eqr 2678	. . . . . 6 |- ( ( ph /\ A = -u B ) -> ( Re ` A ) = -u ( Re ` B ) )
13		sqeqd.4	. . . . . . . . . . . 12 |- ( ph -> 0 <_ ( Re ` A ) )
14	13	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ A = -u B ) -> 0 <_ ( Re ` A ) )
15	14, 12	breqtrd 4679	. . . . . . . . . 10 |- ( ( ph /\ A = -u B ) -> 0 <_ -u ( Re ` B ) )
16	3	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ A = -u B ) -> B e. CC )
17		recl 13850	. . . . . . . . . . . 12 |- ( B e. CC -> ( Re ` B ) e. RR )
18	16, 17	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ A = -u B ) -> ( Re ` B ) e. RR )
19	18	le0neg1d 10599	. . . . . . . . . 10 |- ( ( ph /\ A = -u B ) -> ( ( Re ` B ) <_ 0 <-> 0 <_ -u ( Re ` B ) ) )
20	15, 19	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ A = -u B ) -> ( Re ` B ) <_ 0 )
21		sqeqd.5	. . . . . . . . . 10 |- ( ph -> 0 <_ ( Re ` B ) )
22	21	adantr 481	. . . . . . . . 9 |- ( ( ph /\ A = -u B ) -> 0 <_ ( Re ` B ) )
23		0re 10040	. . . . . . . . . 10 |- 0 e. RR
24		letri3 10123	. . . . . . . . . 10 |- ( ( ( Re ` B ) e. RR /\ 0 e. RR ) -> ( ( Re ` B ) = 0 <-> ( ( Re ` B ) <_ 0 /\ 0 <_ ( Re ` B ) ) ) )
25	18, 23, 24	sylancl 694	. . . . . . . . 9 |- ( ( ph /\ A = -u B ) -> ( ( Re ` B ) = 0 <-> ( ( Re ` B ) <_ 0 /\ 0 <_ ( Re ` B ) ) ) )
26	20, 22, 25	mpbir2and 957	. . . . . . . 8 |- ( ( ph /\ A = -u B ) -> ( Re ` B ) = 0 )
27	26	negeqd 10275	. . . . . . 7 |- ( ( ph /\ A = -u B ) -> -u ( Re ` B ) = -u 0 )
28		neg0 10327	. . . . . . 7 |- -u 0 = 0
29	27, 28	syl6eq 2672	. . . . . 6 |- ( ( ph /\ A = -u B ) -> -u ( Re ` B ) = 0 )
30	12, 29	eqtrd 2656	. . . . 5 |- ( ( ph /\ A = -u B ) -> ( Re ` A ) = 0 )
31		sqeqd.6	. . . . 5 |- ( ( ph /\ ( Re ` A ) = 0 /\ ( Re ` B ) = 0 ) -> A = B )
32	8, 30, 26, 31	syl3anc 1326	. . . 4 |- ( ( ph /\ A = -u B ) -> A = B )
33	32	ex 450	. . 3 |- ( ph -> ( A = -u B -> A = B ) )
34	7, 33	syld 47	. 2 |- ( ph -> ( -. A = B -> A = B ) )
35	34	pm2.18d 124	1 |- ( ph -> A = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   <_ cle 10075   -ucneg 10267   2c2 11070   ^cexp 12860   Recre 13837

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-cnex 9992  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  ax-pre-mulgt0 10013

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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841

This theorem is referenced by: (None)