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Theorem eengv 25859

Description: The value of the Euclidean geometry for dimension

. (Contributed by Thierry Arnoux, 15-Mar-2019.)

**Assertion**
Ref	Expression
eengv	EEG Itv LineG $\$ $\/$ $\/$

Proof of Theorem eengv Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5
2	1	opeq2d 4409	. . . 4
3	1	adantr 481	. . . . . 6 $/\$
4		simpl 473	. . . . . . . 8 $/\$ $/\$
5	4	oveq2d 6666	. . . . . . 7 $/\$ $/\$
6	5	sumeq1d 14431	. . . . . 6 $/\$ $/\$
7	1, 3, 6	mpt2eq123dva 6716	. . . . 5
8	7	opeq2d 4409	. . . 4
9	2, 8	preq12d 4276	. . 3
10	1	adantr 481	. . . . . . 7 $/\$ $/\$
11	10	rabeqdv 3194	. . . . . 6 $/\$ $/\$
12	1, 3, 11	mpt2eq123dva 6716	. . . . 5
13	12	opeq2d 4409	. . . 4 Itv Itv
14	3	difeq1d 3727	. . . . . 6 $/\$ $\$ $\$
15	1	rabeqdv 3194	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
16	15	adantr 481	. . . . . 6 $/\$ $/\$ $\$ $\/$ $\/$ $\/$ $\/$
17	1, 14, 16	mpt2eq123dva 6716	. . . . 5 $\$ $\/$ $\/$ $\$ $\/$ $\/$
18	17	opeq2d 4409	. . . 4 LineG $\$ $\/$ $\/$ LineG $\$ $\/$ $\/$
19	13, 18	preq12d 4276	. . 3 Itv LineG $\$ $\/$ $\/$ Itv LineG $\$ $\/$ $\/$
20	9, 19	uneq12d 3768	. 2 Itv LineG $\$ $\/$ $\/$ Itv LineG $\$ $\/$ $\/$
21		df-eeng 25858	. 2 EEG Itv LineG $\$ $\/$ $\/$
22		prex 4909	. . 3
23		prex 4909	. . 3 Itv LineG $\$ $\/$ $\/$
24	22, 23	unex 6956	. 2 Itv LineG $\$ $\/$ $\/$
25	20, 21, 24	fvmpt 6282	1 EEG Itv LineG $\$ $\/$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990

crab 2916 $\$ cdif 3571

cun 3572

csn 4177

cpr 4179

cop 4183 class class class wbr 4653

cfv 5888 (class class class)co 6650

cmpt2 6652

c1 9937

cmin 10266

cn 11020

c2 11070

cfz 12326

cexp 12860

csu 14416

cnx 15854

cbs 15857

cds 15950 Itvcitv 25335 LineGclng 25336

cee 25768

cbtwn 25769 EEGceeng 25857

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-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 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-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-iota 5851 df-fun 5890 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seq 12802 df-sum 14417 df-eeng 25858

This theorem is referenced by: eengstr 25860 eengbas 25861 ebtwntg 25862 ecgrtg 25863 elntg 25864