ax5seglem9 - Metamath Proof Explorer

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Theorem ax5seglem9 25817

Description: Lemma for ax5seg 25818. Take the calculation in ax5seglem8 25816 and turn it into a series of measurements. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)

**Assertion**
Ref	Expression
ax5seglem9	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Distinct variable groups:

**Proof of Theorem ax5seglem9**
Step	Hyp	Ref	Expression
1		fzfid 12772	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2		0re 10040	. . . . . . 7
3		1re 10039	. . . . . . 7
4	2, 3	elicc2i 12239	. . . . . 6 $/\$ $/\$
5	4	simp1bi 1076	. . . . 5
6	5	recnd 10068	. . . 4
7	6	ad2antrl 764	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simprrl 804	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
9	8	ad2antrr 762	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		fveecn 25782	. . . . . 6 $/\$
11	9, 10	sylancom 701	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simprrr 805	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
13	12	ad2antrr 762	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		fveecn 25782	. . . . . 6 $/\$
15	13, 14	sylancom 701	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	11, 15	subcld 10392	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17	16	sqcld 13006	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	1, 7, 17	fsummulc2 14516	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		simprll 802	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
20	19	ad2antrr 762	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		fveecn 25782	. . . . . . . . . . . . 13 $/\$
22	20, 21	sylancom 701	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	22, 11	subcld 10392	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	23	sqcld 13006	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	1, 7, 24	fsummulc2 14516	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	25	oveq1d 6665	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	7	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	27, 24	mulcld 10060	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	22, 15	subcld 10392	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29	sqcld 13006	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	1, 28, 30	fsumsub 14520	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	26, 31	eqtr4d 2659	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	oveq2d 6666	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34		ax-1cn 9994	. . . . . . . 8
35		subcl 10280	. . . . . . . 8 $/\$
36	34, 7, 35	sylancr 695	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	28, 30	subcld 10392	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	1, 36, 37	fsummulc2 14516	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	33, 38	eqtrd 2656	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	oveq2d 6666	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		simprlr 803	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
42	41	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		fveecn 25782	. . . . . . . 8 $/\$
44	42, 43	sylancom 701	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	44, 15	subcld 10392	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	45	sqcld 13006	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	34, 27, 35	sylancr 695	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	47, 37	mulcld 10060	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	1, 46, 48	fsumadd 14470	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	40, 49	eqtr4d 2659	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51		fveq2 6191	. . . . . . . . 9
52		fveq2 6191	. . . . . . . . . . 11
53	52	oveq2d 6666	. . . . . . . . . 10
54		fveq2 6191	. . . . . . . . . . 11
55	54	oveq2d 6666	. . . . . . . . . 10
56	53, 55	oveq12d 6668	. . . . . . . . 9
57	51, 56	eqeq12d 2637	. . . . . . . 8
58	57	rspccva 3308	. . . . . . 7 $/\$
59	58	adantll 750	. . . . . 6 $/\$ $/\$
60	59	adantll 750	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61		ax5seglem8 25816	. . . . . . 7 $/\$ $/\$ $/\$
62		oveq1 6657	. . . . . . . . . 10
63	62	oveq1d 6665	. . . . . . . . 9
64	63	oveq1d 6665	. . . . . . . 8
65	64	eqcomd 2628	. . . . . . 7
66	61, 65	sylan9eq 2676	. . . . . 6 $/\$ $/\$ $/\$ $/\$
67	66	3impa 1259	. . . . 5 $/\$ $/\$ $/\$ $/\$
68	22, 27, 11, 15, 60, 67	syl221anc 1337	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68	sumeq2dv 14433	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	50, 69	eqtr4d 2659	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	18, 70	eqtr4d 2659	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

cle 10075

cmin 10266

cn 11020

c2 11070

cicc 12178

cfz 12326

cexp 12860

csu 14416

cee 25768

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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 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-fal 1489 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-int 4476 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-se 5074 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-isom 5897 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-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 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-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-ee 25771

This theorem is referenced by: ax5seg 25818

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