Theorem mulcnsrec 9542

Description: Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 7395, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 9540.

Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 9243. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
mulcnsrec

Proof of Theorem mulcnsrec

Step	Hyp	Ref	Expression
1		mulcnsr 9534	. 2
2		opex 4716	. . . 4
3	2	ecid 7395	. . 3
4		opex 4716	. . . 4
5	4	ecid 7395	. . 3
6	3, 5	oveq12i 6308	. 2
7		opex 4716	. . 3
8	7	ecid 7395	. 2
9	1, 6, 8	3eqtr4g 2523	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  <.cop 4035  cep 4794  `'ccnv 5003  (class class class)co 6296  [cec 7328  cnr 9264  cm1r 9267  cplr 9268  cmr 9269  cmul 9518

This theorem is referenced by:  axmulcom  9553  axmulass  9555  axdistr  9556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-eprel 4796  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-ec 7332  df-c 9519  df-mul 9525