Theorem plusfreseq 41772

Description: If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily 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
plusfreseq	|- ( (/) e/ ran .+^ -> ( .+ |` ( B X. B ) ) = .+^ )

Proof of Theorem plusfreseq

Dummy variables x p y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plusfreseq.1	. . . . 5 |- B = ( Base ` M )
2		plusfreseq.3	. . . . 5 |- .+^ = ( +f ` M )
3	1, 2	plusffn 17250	. . . 4 |- .+^ Fn ( B X. B )
4		fnfun 5988	. . . 4 |- ( .+^ Fn ( B X. B ) -> Fun .+^ )
5	3, 4	ax-mp 5	. . 3 |- Fun .+^
6	5	a1i 11	. 2 |- ( (/) e/ ran .+^ -> Fun .+^ )
7		id 22	. 2 |- ( (/) e/ ran .+^ -> (/) e/ ran .+^ )
8		plusfreseq.2	. . . . . . 7 |- .+ = ( +g ` M )
9	1, 8, 2	plusfval 17248	. . . . . 6 |- ( ( x e. B /\ y e. B ) -> ( x .+^ y ) = ( x .+ y ) )
10	9	eqcomd 2628	. . . . 5 |- ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( x .+^ y ) )
11	10	rgen2a 2977	. . . 4 |- A. x e. B A. y e. B ( x .+ y ) = ( x .+^ y )
12	11	a1i 11	. . 3 |- ( (/) e/ ran .+^ -> A. x e. B A. y e. B ( x .+ y ) = ( x .+^ y ) )
13		fveq2 6191	. . . . . 6 |- ( p = <. x , y >. -> ( .+ ` p ) = ( .+ ` <. x , y >. ) )
14		df-ov 6653	. . . . . 6 |- ( x .+ y ) = ( .+ ` <. x , y >. )
15	13, 14	syl6eqr 2674	. . . . 5 |- ( p = <. x , y >. -> ( .+ ` p ) = ( x .+ y ) )
16		fveq2 6191	. . . . . 6 |- ( p = <. x , y >. -> ( .+^ ` p ) = ( .+^ ` <. x , y >. ) )
17		df-ov 6653	. . . . . 6 |- ( x .+^ y ) = ( .+^ ` <. x , y >. )
18	16, 17	syl6eqr 2674	. . . . 5 |- ( p = <. x , y >. -> ( .+^ ` p ) = ( x .+^ y ) )
19	15, 18	eqeq12d 2637	. . . 4 |- ( p = <. x , y >. -> ( ( .+ ` p ) = ( .+^ ` p ) <-> ( x .+ y ) = ( x .+^ y ) ) )
20	19	ralxp 5263	. . 3 |- ( A. p e. ( B X. B ) ( .+ ` p ) = ( .+^ ` p ) <-> A. x e. B A. y e. B ( x .+ y ) = ( x .+^ y ) )
21	12, 20	sylibr 224	. 2 |- ( (/) e/ ran .+^ -> A. p e. ( B X. B ) ( .+ ` p ) = ( .+^ ` p ) )
22		fndm 5990	. . . . 5 |- ( .+^ Fn ( B X. B ) -> dom .+^ = ( B X. B ) )
23	22	eqcomd 2628	. . . 4 |- ( .+^ Fn ( B X. B ) -> ( B X. B ) = dom .+^ )
24	3, 23	ax-mp 5	. . 3 |- ( B X. B ) = dom .+^
25	24	fveqressseq 6355	. 2 |- ( ( Fun .+^ /\ (/) e/ ran .+^ /\ A. p e. ( B X. B ) ( .+ ` p ) = ( .+^ ` p ) ) -> ( .+ |` ( B X. B ) ) = .+^ )
26	6, 7, 21, 25	syl3anc 1326	1 |- ( (/) e/ ran .+^ -> ( .+ |` ( B X. B ) ) = .+^ )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   A.wral 2912   (/)c0 3915   <.cop 4183   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   +fcplusf 17239

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

This theorem is referenced by:  mgmplusfreseq  41773