Theorem qqhre 30064

Description: The QQHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)

Assertion

Ref	Expression
qqhre	|- (QQHom ` RRfld ) = ( _I |` QQ )

Proof of Theorem qqhre

Step	Hyp	Ref	Expression
1		resubdrg 19954	. . . . . . 7 |- ( RR e. (SubRing ` ℂfld ) /\ RRfld e. DivRing )
2	1	simpri 478	. . . . . 6 |- RRfld e. DivRing
3		drngring 18754	. . . . . . 7 |- (RRfld e. DivRing -> RRfld e. Ring )
4		f1oi 6174	. . . . . . . . . . 11 |- ( _I |` ZZ ) : ZZ -1-1-onto-> ZZ
5		f1of1 6136	. . . . . . . . . . 11 |- ( ( _I |` ZZ ) : ZZ -1-1-onto-> ZZ -> ( _I |` ZZ ) : ZZ -1-1-> ZZ )
6	4, 5	ax-mp 5	. . . . . . . . . 10 |- ( _I |` ZZ ) : ZZ -1-1-> ZZ
7		zssre 11384	. . . . . . . . . 10 |- ZZ C_ RR
8		f1ss 6106	. . . . . . . . . 10 |- ( ( ( _I |` ZZ ) : ZZ -1-1-> ZZ /\ ZZ C_ RR ) -> ( _I |` ZZ ) : ZZ -1-1-> RR )
9	6, 7, 8	mp2an 708	. . . . . . . . 9 |- ( _I |` ZZ ) : ZZ -1-1-> RR
10		zrhre 30063	. . . . . . . . . 10 |- ( ZRHom ` RRfld ) = ( _I |` ZZ )
11		f1eq1 6096	. . . . . . . . . 10 |- ( ( ZRHom ` RRfld ) = ( _I |` ZZ ) -> ( ( ZRHom ` RRfld ) : ZZ -1-1-> RR <-> ( _I |` ZZ ) : ZZ -1-1-> RR ) )
12	10, 11	ax-mp 5	. . . . . . . . 9 |- ( ( ZRHom ` RRfld ) : ZZ -1-1-> RR <-> ( _I |` ZZ ) : ZZ -1-1-> RR )
13	9, 12	mpbir 221	. . . . . . . 8 |- ( ZRHom ` RRfld ) : ZZ -1-1-> RR
14		rebase 19952	. . . . . . . . 9 |- RR = ( Base ` RRfld )
15		eqid 2622	. . . . . . . . 9 |- ( ZRHom ` RRfld ) = ( ZRHom ` RRfld )
16		re0g 19958	. . . . . . . . 9 |- 0 = ( 0g ` RRfld )
17	14, 15, 16	zrhchr 30020	. . . . . . . 8 |- (RRfld e. Ring -> ( (chr ` RRfld ) = 0 <-> ( ZRHom ` RRfld ) : ZZ -1-1-> RR ) )
18	13, 17	mpbiri 248	. . . . . . 7 |- (RRfld e. Ring -> (chr ` RRfld ) = 0 )
19	2, 3, 18	mp2b 10	. . . . . 6 |- (chr ` RRfld ) = 0
20		eqid 2622	. . . . . . 7 |- (/r ` RRfld ) = (/r ` RRfld )
21	14, 20, 15	qqhf 30030	. . . . . 6 |- ( (RRfld e. DivRing /\ (chr ` RRfld ) = 0 ) -> (QQHom ` RRfld ) : QQ --> RR )
22	2, 19, 21	mp2an 708	. . . . 5 |- (QQHom ` RRfld ) : QQ --> RR
23	22	a1i 11	. . . 4 |- ( T. -> (QQHom ` RRfld ) : QQ --> RR )
24	23	feqmptd 6249	. . 3 |- ( T. -> (QQHom ` RRfld ) = ( q e. QQ |-> ( (QQHom ` RRfld ) ` q ) ) )
25	24	trud 1493	. 2 |- (QQHom ` RRfld ) = ( q e. QQ |-> ( (QQHom ` RRfld ) ` q ) )
26	14, 20, 15	qqhvval 30027	. . . . 5 |- ( ( (RRfld e. DivRing /\ (chr ` RRfld ) = 0 ) /\ q e. QQ ) -> ( (QQHom ` RRfld ) ` q ) = ( ( ( ZRHom ` RRfld ) ` (numer ` q ) ) (/r ` RRfld ) ( ( ZRHom ` RRfld ) ` (denom ` q ) ) ) )
27	2, 19, 26	mpanl12 718	. . . 4 |- ( q e. QQ -> ( (QQHom ` RRfld ) ` q ) = ( ( ( ZRHom ` RRfld ) ` (numer ` q ) ) (/r ` RRfld ) ( ( ZRHom ` RRfld ) ` (denom ` q ) ) ) )
28		f1f 6101	. . . . . . . 8 |- ( ( ZRHom ` RRfld ) : ZZ -1-1-> RR -> ( ZRHom ` RRfld ) : ZZ --> RR )
29	13, 28	ax-mp 5	. . . . . . 7 |- ( ZRHom ` RRfld ) : ZZ --> RR
30	29	a1i 11	. . . . . 6 |- ( q e. QQ -> ( ZRHom ` RRfld ) : ZZ --> RR )
31		qnumcl 15448	. . . . . 6 |- ( q e. QQ -> (numer ` q ) e. ZZ )
32	30, 31	ffvelrnd 6360	. . . . 5 |- ( q e. QQ -> ( ( ZRHom ` RRfld ) ` (numer ` q ) ) e. RR )
33		qdencl 15449	. . . . . . 7 |- ( q e. QQ -> (denom ` q ) e. NN )
34	33	nnzd 11481	. . . . . 6 |- ( q e. QQ -> (denom ` q ) e. ZZ )
35	30, 34	ffvelrnd 6360	. . . . 5 |- ( q e. QQ -> ( ( ZRHom ` RRfld ) ` (denom ` q ) ) e. RR )
36	34	anim1i 592	. . . . . . . 8 |- ( ( q e. QQ /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = 0 ) -> ( (denom ` q ) e. ZZ /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = 0 ) )
37	14, 15, 16	zrhf1ker 30019	. . . . . . . . . . . 12 |- (RRfld e. Ring -> ( ( ZRHom ` RRfld ) : ZZ -1-1-> RR <-> ( `' ( ZRHom ` RRfld ) " { 0 } ) = { 0 } ) )
38	2, 3, 37	mp2b 10	. . . . . . . . . . 11 |- ( ( ZRHom ` RRfld ) : ZZ -1-1-> RR <-> ( `' ( ZRHom ` RRfld ) " { 0 } ) = { 0 } )
39	13, 38	mpbi 220	. . . . . . . . . 10 |- ( `' ( ZRHom ` RRfld ) " { 0 } ) = { 0 }
40	39	eleq2i 2693	. . . . . . . . 9 |- ( (denom ` q ) e. ( `' ( ZRHom ` RRfld ) " { 0 } ) <-> (denom ` q ) e. { 0 } )
41		ffn 6045	. . . . . . . . . 10 |- ( ( ZRHom ` RRfld ) : ZZ --> RR -> ( ZRHom ` RRfld ) Fn ZZ )
42		fniniseg 6338	. . . . . . . . . 10 |- ( ( ZRHom ` RRfld ) Fn ZZ -> ( (denom ` q ) e. ( `' ( ZRHom ` RRfld ) " { 0 } ) <-> ( (denom ` q ) e. ZZ /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = 0 ) ) )
43	29, 41, 42	mp2b 10	. . . . . . . . 9 |- ( (denom ` q ) e. ( `' ( ZRHom ` RRfld ) " { 0 } ) <-> ( (denom ` q ) e. ZZ /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = 0 ) )
44		fvex 6201	. . . . . . . . . 10 |- (denom ` q ) e. _V
45	44	elsn 4192	. . . . . . . . 9 |- ( (denom ` q ) e. { 0 } <-> (denom ` q ) = 0 )
46	40, 43, 45	3bitr3ri 291	. . . . . . . 8 |- ( (denom ` q ) = 0 <-> ( (denom ` q ) e. ZZ /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = 0 ) )
47	36, 46	sylibr 224	. . . . . . 7 |- ( ( q e. QQ /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = 0 ) -> (denom ` q ) = 0 )
48	33	nnne0d 11065	. . . . . . . . 9 |- ( q e. QQ -> (denom ` q ) =/= 0 )
49	48	adantr 481	. . . . . . . 8 |- ( ( q e. QQ /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = 0 ) -> (denom ` q ) =/= 0 )
50	49	neneqd 2799	. . . . . . 7 |- ( ( q e. QQ /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = 0 ) -> -. (denom ` q ) = 0 )
51	47, 50	pm2.65da 600	. . . . . 6 |- ( q e. QQ -> -. ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = 0 )
52	51	neqned 2801	. . . . 5 |- ( q e. QQ -> ( ( ZRHom ` RRfld ) ` (denom ` q ) ) =/= 0 )
53		redvr 19963	. . . . 5 |- ( ( ( ( ZRHom ` RRfld ) ` (numer ` q ) ) e. RR /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) e. RR /\ ( ( ZRHom ` RRfld ) ` (denom ` q ) ) =/= 0 ) -> ( ( ( ZRHom ` RRfld ) ` (numer ` q ) ) (/r ` RRfld ) ( ( ZRHom ` RRfld ) ` (denom ` q ) ) ) = ( ( ( ZRHom ` RRfld ) ` (numer ` q ) ) / ( ( ZRHom ` RRfld ) ` (denom ` q ) ) ) )
54	32, 35, 52, 53	syl3anc 1326	. . . 4 |- ( q e. QQ -> ( ( ( ZRHom ` RRfld ) ` (numer ` q ) ) (/r ` RRfld ) ( ( ZRHom ` RRfld ) ` (denom ` q ) ) ) = ( ( ( ZRHom ` RRfld ) ` (numer ` q ) ) / ( ( ZRHom ` RRfld ) ` (denom ` q ) ) ) )
55	10	fveq1i 6192	. . . . . . . 8 |- ( ( ZRHom ` RRfld ) ` (numer ` q ) ) = ( ( _I |` ZZ ) ` (numer ` q ) )
56		fvresi 6439	. . . . . . . 8 |- ( (numer ` q ) e. ZZ -> ( ( _I |` ZZ ) ` (numer ` q ) ) = (numer ` q ) )
57	55, 56	syl5eq 2668	. . . . . . 7 |- ( (numer ` q ) e. ZZ -> ( ( ZRHom ` RRfld ) ` (numer ` q ) ) = (numer ` q ) )
58	31, 57	syl 17	. . . . . 6 |- ( q e. QQ -> ( ( ZRHom ` RRfld ) ` (numer ` q ) ) = (numer ` q ) )
59	10	fveq1i 6192	. . . . . . . 8 |- ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = ( ( _I |` ZZ ) ` (denom ` q ) )
60		fvresi 6439	. . . . . . . 8 |- ( (denom ` q ) e. ZZ -> ( ( _I |` ZZ ) ` (denom ` q ) ) = (denom ` q ) )
61	59, 60	syl5eq 2668	. . . . . . 7 |- ( (denom ` q ) e. ZZ -> ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = (denom ` q ) )
62	34, 61	syl 17	. . . . . 6 |- ( q e. QQ -> ( ( ZRHom ` RRfld ) ` (denom ` q ) ) = (denom ` q ) )
63	58, 62	oveq12d 6668	. . . . 5 |- ( q e. QQ -> ( ( ( ZRHom ` RRfld ) ` (numer ` q ) ) / ( ( ZRHom ` RRfld ) ` (denom ` q ) ) ) = ( (numer ` q ) / (denom ` q ) ) )
64		qeqnumdivden 15454	. . . . 5 |- ( q e. QQ -> q = ( (numer ` q ) / (denom ` q ) ) )
65	63, 64	eqtr4d 2659	. . . 4 |- ( q e. QQ -> ( ( ( ZRHom ` RRfld ) ` (numer ` q ) ) / ( ( ZRHom ` RRfld ) ` (denom ` q ) ) ) = q )
66	27, 54, 65	3eqtrd 2660	. . 3 |- ( q e. QQ -> ( (QQHom ` RRfld ) ` q ) = q )
67	66	mpteq2ia 4740	. 2 |- ( q e. QQ |-> ( (QQHom ` RRfld ) ` q ) ) = ( q e. QQ |-> q )
68		mptresid 5456	. 2 |- ( q e. QQ |-> q ) = ( _I |` QQ )
69	25, 67, 68	3eqtri 2648	1 |- (QQHom ` RRfld ) = ( _I |` QQ )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   C_ wss 3574   {csn 4177   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   / cdiv 10684   ZZcz 11377   QQcq 11788  numercnumer 15441  denomcdenom 15442   Ringcrg 18547  /_rcdvr 18682   DivRingcdr 18747  SubRingcsubrg 18776  ℂ_fldccnfld 19746   ZRHomczrh 19848  chrcchr 19850  RRfldcrefld 19950  QQHomcqqh 30016

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444  df-gz 15634  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-od 17948  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-chr 19854  df-refld 19951  df-qqh 30017

This theorem is referenced by:  rrhre  30065