Theorem ofldtos 29811

Description: An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)

Assertion

Ref	Expression
ofldtos	|- ( F e. oField -> F e. Toset )

Proof of Theorem ofldtos

Step	Hyp	Ref	Expression
1		isofld 29802	. . . 4 |- ( F e. oField <-> ( F e. Field /\ F e. oRing ) )
2	1	simprbi 480	. . 3 |- ( F e. oField -> F e. oRing )
3		orngogrp 29801	. . 3 |- ( F e. oRing -> F e. oGrp )
4		isogrp 29702	. . . 4 |- ( F e. oGrp <-> ( F e. Grp /\ F e. oMnd ) )
5	4	simprbi 480	. . 3 |- ( F e. oGrp -> F e. oMnd )
6	2, 3, 5	3syl 18	. 2 |- ( F e. oField -> F e. oMnd )
7		omndtos 29705	. 2 |- ( F e. oMnd -> F e. Toset )
8	6, 7	syl 17	1 |- ( F e. oField -> F e. Toset )

Syntax hints:   -> wi 4   e. wcel 1990  Tosetctos 17033   Grpcgrp 17422  Fieldcfield 18748  oMndcomnd 29697  oGrpcogrp 29698  oRingcorng 29795  oFieldcofld 29796

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

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-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-iota 5851  df-fv 5896  df-ov 6653  df-omnd 29699  df-ogrp 29700  df-orng 29797  df-ofld 29798

This theorem is referenced by:  ofldchr  29814