Theorem mgmplusfreseq 41773

Description: If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)

Hypotheses

Ref	Expression
plusfreseq.1	|- B = ( Base ` M )
plusfreseq.2	|- .+ = ( +g ` M )
plusfreseq.3	|- .+^ = ( +f ` M )

Assertion

Ref	Expression
mgmplusfreseq	|- ( ( M e. Mgm /\ (/) e/ B ) -> ( .+ |` ( B X. B ) ) = .+^ )

Proof of Theorem mgmplusfreseq

Step	Hyp	Ref	Expression
1		plusfreseq.1	. . . . 5 |- B = ( Base ` M )
2		plusfreseq.3	. . . . 5 |- .+^ = ( +f ` M )
3	1, 2	mgmplusf 17251	. . . 4 |- ( M e. Mgm -> .+^ : ( B X. B ) --> B )
4		frn 6053	. . . 4 |- ( .+^ : ( B X. B ) --> B -> ran .+^ C_ B )
5		ssel 3597	. . . . 5 |- ( ran .+^ C_ B -> ( (/) e. ran .+^ -> (/) e. B ) )
6	5	nelcon3d 2909	. . . 4 |- ( ran .+^ C_ B -> ( (/) e/ B -> (/) e/ ran .+^ ) )
7	3, 4, 6	3syl 18	. . 3 |- ( M e. Mgm -> ( (/) e/ B -> (/) e/ ran .+^ ) )
8	7	imp 445	. 2 |- ( ( M e. Mgm /\ (/) e/ B ) -> (/) e/ ran .+^ )
9		plusfreseq.2	. . 3 |- .+ = ( +g ` M )
10	1, 9, 2	plusfreseq 41772	. 2 |- ( (/) e/ ran .+^ -> ( .+ |` ( B X. B ) ) = .+^ )
11	8, 10	syl 17	1 |- ( ( M e. Mgm /\ (/) e/ B ) -> ( .+ |` ( B X. B ) ) = .+^ )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   C_ wss 3574   (/)c0 3915   X. cxp 5112   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888   Basecbs 15857   +g cplusg 15941   +fcplusf 17239  Mgmcmgm 17240

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

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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-plusf 17241  df-mgm 17242

This theorem is referenced by: (None)