Theorem xmulasslem 12115

Description: Lemma for xmulass 12117. (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
xmulasslem.1	|- ( x = D -> ( ps <-> X = Y ) )
xmulasslem.2	|- ( x = -e D -> ( ps <-> E = F ) )
xmulasslem.x	|- ( ph -> X e. RR* )
xmulasslem.y	|- ( ph -> Y e. RR* )
xmulasslem.d	|- ( ph -> D e. RR* )
xmulasslem.ps	|- ( ( ph /\ ( x e. RR* /\ 0 < x ) ) -> ps )
xmulasslem.0	|- ( ph -> ( x = 0 -> ps ) )
xmulasslem.e	|- ( ph -> E = -e X )
xmulasslem.f	|- ( ph -> F = -e Y )

Assertion

Ref	Expression
xmulasslem	|- ( ph -> X = Y )

Distinct variable groups:   x, D   x, E   x, F   ph, x   x, X   x, Y

Allowed substitution hint:   ps( x)

Proof of Theorem xmulasslem

Step	Hyp	Ref	Expression
1		xmulasslem.d	. . 3 |- ( ph -> D e. RR* )
2		0xr 10086	. . 3 |- 0 e. RR*
3		xrltso 11974	. . . 4 |- < Or RR*
4		solin 5058	. . . 4 |- ( ( < Or RR* /\ ( D e. RR* /\ 0 e. RR* ) ) -> ( D < 0 \/ D = 0 \/ 0 < D ) )
5	3, 4	mpan 706	. . 3 |- ( ( D e. RR* /\ 0 e. RR* ) -> ( D < 0 \/ D = 0 \/ 0 < D ) )
6	1, 2, 5	sylancl 694	. 2 |- ( ph -> ( D < 0 \/ D = 0 \/ 0 < D ) )
7		xlt0neg1 12050	. . . . . 6 |- ( D e. RR* -> ( D < 0 <-> 0 < -e D ) )
8	1, 7	syl 17	. . . . 5 |- ( ph -> ( D < 0 <-> 0 < -e D ) )
9		xnegcl 12044	. . . . . . 7 |- ( D e. RR* -> -e D e. RR* )
10	1, 9	syl 17	. . . . . 6 |- ( ph -> -e D e. RR* )
11		breq2 4657	. . . . . . . . 9 |- ( x = -e D -> ( 0 < x <-> 0 < -e D ) )
12		xmulasslem.2	. . . . . . . . 9 |- ( x = -e D -> ( ps <-> E = F ) )
13	11, 12	imbi12d 334	. . . . . . . 8 |- ( x = -e D -> ( ( 0 < x -> ps ) <-> ( 0 < -e D -> E = F ) ) )
14	13	imbi2d 330	. . . . . . 7 |- ( x = -e D -> ( ( ph -> ( 0 < x -> ps ) ) <-> ( ph -> ( 0 < -e D -> E = F ) ) ) )
15		xmulasslem.ps	. . . . . . . . 9 |- ( ( ph /\ ( x e. RR* /\ 0 < x ) ) -> ps )
16	15	exp32 631	. . . . . . . 8 |- ( ph -> ( x e. RR* -> ( 0 < x -> ps ) ) )
17	16	com12 32	. . . . . . 7 |- ( x e. RR* -> ( ph -> ( 0 < x -> ps ) ) )
18	14, 17	vtoclga 3272	. . . . . 6 |- ( -e D e. RR* -> ( ph -> ( 0 < -e D -> E = F ) ) )
19	10, 18	mpcom 38	. . . . 5 |- ( ph -> ( 0 < -e D -> E = F ) )
20	8, 19	sylbid 230	. . . 4 |- ( ph -> ( D < 0 -> E = F ) )
21		xmulasslem.e	. . . . . 6 |- ( ph -> E = -e X )
22		xmulasslem.f	. . . . . 6 |- ( ph -> F = -e Y )
23	21, 22	eqeq12d 2637	. . . . 5 |- ( ph -> ( E = F <-> -e X = -e Y ) )
24		xmulasslem.x	. . . . . 6 |- ( ph -> X e. RR* )
25		xmulasslem.y	. . . . . 6 |- ( ph -> Y e. RR* )
26		xneg11 12046	. . . . . 6 |- ( ( X e. RR* /\ Y e. RR* ) -> ( -e X = -e Y <-> X = Y ) )
27	24, 25, 26	syl2anc 693	. . . . 5 |- ( ph -> ( -e X = -e Y <-> X = Y ) )
28	23, 27	bitrd 268	. . . 4 |- ( ph -> ( E = F <-> X = Y ) )
29	20, 28	sylibd 229	. . 3 |- ( ph -> ( D < 0 -> X = Y ) )
30		eqeq1 2626	. . . . . . 7 |- ( x = D -> ( x = 0 <-> D = 0 ) )
31		xmulasslem.1	. . . . . . 7 |- ( x = D -> ( ps <-> X = Y ) )
32	30, 31	imbi12d 334	. . . . . 6 |- ( x = D -> ( ( x = 0 -> ps ) <-> ( D = 0 -> X = Y ) ) )
33	32	imbi2d 330	. . . . 5 |- ( x = D -> ( ( ph -> ( x = 0 -> ps ) ) <-> ( ph -> ( D = 0 -> X = Y ) ) ) )
34		xmulasslem.0	. . . . 5 |- ( ph -> ( x = 0 -> ps ) )
35	33, 34	vtoclg 3266	. . . 4 |- ( D e. RR* -> ( ph -> ( D = 0 -> X = Y ) ) )
36	1, 35	mpcom 38	. . 3 |- ( ph -> ( D = 0 -> X = Y ) )
37		breq2 4657	. . . . . . 7 |- ( x = D -> ( 0 < x <-> 0 < D ) )
38	37, 31	imbi12d 334	. . . . . 6 |- ( x = D -> ( ( 0 < x -> ps ) <-> ( 0 < D -> X = Y ) ) )
39	38	imbi2d 330	. . . . 5 |- ( x = D -> ( ( ph -> ( 0 < x -> ps ) ) <-> ( ph -> ( 0 < D -> X = Y ) ) ) )
40	39, 17	vtoclga 3272	. . . 4 |- ( D e. RR* -> ( ph -> ( 0 < D -> X = Y ) ) )
41	1, 40	mpcom 38	. . 3 |- ( ph -> ( 0 < D -> X = Y ) )
42	29, 36, 41	3jaod 1392	. 2 |- ( ph -> ( ( D < 0 \/ D = 0 \/ 0 < D ) -> X = Y ) )
43	6, 42	mpd 15	1 |- ( ph -> X = Y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   class class class wbr 4653   Or wor 5034   0cc0 9936   RR*cxr 10073   < clt 10074   -ecxne 11943

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

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-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-xneg 11946

This theorem is referenced by:  xmulass  12117