cnre2csqlem - Mathbox for Thierry Arnoux

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Theorem cnre2csqlem 29956

Description: Lemma for cnre2csqima 29957. (Contributed by Thierry Arnoux, 27-Sep-2017.)

**Hypotheses**
Ref	Expression
cnre2csqlem.1
cnre2csqlem.2
cnre2csqlem.3
cnre2csqlem.4
cnre2csqlem.5	$/\$

**Assertion**
Ref	Expression
cnre2csqlem	$/\$ $/\$

Distinct variable groups:

Allowed substitution hints:

(

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(

)

**Proof of Theorem cnre2csqlem**
Step	Hyp	Ref	Expression
1		cnre2csqlem.3	. . . . . . 7
2		ssv 3625	. . . . . . 7
3		fnssres 6004	. . . . . . 7 $/\$
4	1, 2, 3	mp2an 708	. . . . . 6
5		elpreima 6337	. . . . . 6 $/\$
6	4, 5	mp1i 13	. . . . 5 $/\$ $/\$ $/\$
7	6	simplbda 654	. . . 4 $/\$ $/\$ $/\$
8	7	ex 450	. . 3 $/\$ $/\$
9		simp2 1062	. . . . . 6 $/\$ $/\$
10		fvres 6207	. . . . . 6
11	9, 10	syl 17	. . . . 5 $/\$ $/\$
12	11	eleq1d 2686	. . . 4 $/\$ $/\$
13		simp1 1061	. . . . . . . . . . . 12 $/\$ $/\$
14		fveq2 6191	. . . . . . . . . . . . . 14
15	14	eleq1d 2686	. . . . . . . . . . . . 13
16		cnre2csqlem.4	. . . . . . . . . . . . 13
17	15, 16	vtoclga 3272	. . . . . . . . . . . 12
18	13, 17	syl 17	. . . . . . . . . . 11 $/\$ $/\$
19		simp3 1063	. . . . . . . . . . . 12 $/\$ $/\$
20	19	rpred 11872	. . . . . . . . . . 11 $/\$ $/\$
21	18, 20	resubcld 10458	. . . . . . . . . 10 $/\$ $/\$
22	21	rexrd 10089	. . . . . . . . 9 $/\$ $/\$
23	18, 20	readdcld 10069	. . . . . . . . . 10 $/\$ $/\$
24	23	rexrd 10089	. . . . . . . . 9 $/\$ $/\$
25		elioo2 12216	. . . . . . . . 9 $/\$ $/\$ $/\$
26	22, 24, 25	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
27	26	biimpa 501	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
28	27	simp2d 1074	. . . . . 6 $/\$ $/\$ $/\$
29	27	simp3d 1075	. . . . . 6 $/\$ $/\$ $/\$
30	28, 29	jca 554	. . . . 5 $/\$ $/\$ $/\$ $/\$
31	30	ex 450	. . . 4 $/\$ $/\$ $/\$
32	12, 31	sylbid 230	. . 3 $/\$ $/\$ $/\$
33		fveq2 6191	. . . . . . 7
34	33	eleq1d 2686	. . . . . 6
35	34, 16	vtoclga 3272	. . . . 5
36	9, 35	syl 17	. . . 4 $/\$ $/\$
37		absdiflt 14057	. . . . 5 $/\$ $/\$ $/\$
38	37	biimprd 238	. . . 4 $/\$ $/\$ $/\$
39	36, 18, 20, 38	syl3anc 1326	. . 3 $/\$ $/\$ $/\$
40	8, 32, 39	3syld 60	. 2 $/\$ $/\$
41		cnre2csqlem.2	. . . . . . 7
42		fnfvelrn 6356	. . . . . . 7 $/\$
43	41, 9, 42	sylancr 695	. . . . . 6 $/\$ $/\$
44		fnfvelrn 6356	. . . . . . 7 $/\$
45	41, 13, 44	sylancr 695	. . . . . 6 $/\$ $/\$
46		oveq1 6657	. . . . . . . . 9
47	46	fveq2d 6195	. . . . . . . 8
48		fveq2 6191	. . . . . . . . 9
49	48	oveq1d 6665	. . . . . . . 8
50	47, 49	eqeq12d 2637	. . . . . . 7
51		oveq2 6658	. . . . . . . . 9
52	51	fveq2d 6195	. . . . . . . 8
53		fveq2 6191	. . . . . . . . 9
54	53	oveq2d 6666	. . . . . . . 8
55	52, 54	eqeq12d 2637	. . . . . . 7
56		cnre2csqlem.5	. . . . . . 7 $/\$
57	50, 55, 56	vtocl2ga 3274	. . . . . 6 $/\$
58	43, 45, 57	syl2anc 693	. . . . 5 $/\$ $/\$
59		cnre2csqlem.1	. . . . . . . 8
60	59	fveq1i 6192	. . . . . . 7
61		fvco2 6273	. . . . . . . 8 $/\$
62	41, 9, 61	sylancr 695	. . . . . . 7 $/\$ $/\$
63	60, 11, 62	3eqtr3a 2680	. . . . . 6 $/\$ $/\$
64	59	fveq1i 6192	. . . . . . 7
65		fvres 6207	. . . . . . . 8
66	13, 65	syl 17	. . . . . . 7 $/\$ $/\$
67		fvco2 6273	. . . . . . . 8 $/\$
68	41, 13, 67	sylancr 695	. . . . . . 7 $/\$ $/\$
69	64, 66, 68	3eqtr3a 2680	. . . . . 6 $/\$ $/\$
70	63, 69	oveq12d 6668	. . . . 5 $/\$ $/\$
71	58, 70	eqtr4d 2659	. . . 4 $/\$ $/\$
72	71	fveq2d 6195	. . 3 $/\$ $/\$
73	72	breq1d 4663	. 2 $/\$ $/\$
74	40, 73	sylibrd 249	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cvv 3200

wss 3574 class class class wbr 4653

cxp 5112

ccnv 5113

crn 5115

cres 5116

cima 5117

ccom 5118

wfn 5883

cfv 5888 (class class class)co 6650

cr 9935

caddc 9939

cxr 10073

clt 10074

cmin 10266

crp 11832

cioo 12175

cabs 13974

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 ax-pre-sup 10014

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-1st 7168 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-sup 8348 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-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ioo 12179 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: cnre2csqima 29957

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