qqhre - Mathbox for Thierry Arnoux

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Theorem qqhre 34327

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

**Assertion**
Ref	Expression
qqhre	⊢ (ℚHom‘ℝ_fld) = ( I ↾ ℚ)

**Proof of Theorem qqhre**
Step	Hyp	Ref	Expression
1		resubdrg 21718	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
2	1	simpri 490	. . . . 5 ⊢ ℝ_fld ∈ DivRing
3		drngring 20811	. . . . . 6 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
4		f1oi 6849	. . . . . . . . . 10 ⊢ ( I ↾ ℤ):ℤ–1-1-onto→ℤ
5		f1of1 6809	. . . . . . . . . 10 ⊢ (( I ↾ ℤ):ℤ–1-1-onto→ℤ → ( I ↾ ℤ):ℤ–1-1→ℤ)
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ( I ↾ ℤ):ℤ–1-1→ℤ
7		zssre 12589	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
8		f1ss 6771	. . . . . . . . 9 ⊢ ((( I ↾ ℤ):ℤ–1-1→ℤ ∧ ℤ ⊆ ℝ) → ( I ↾ ℤ):ℤ–1-1→ℝ)
9	6, 7, 8	mp2an 704	. . . . . . . 8 ⊢ ( I ↾ ℤ):ℤ–1-1→ℝ
10		zrhre 34326	. . . . . . . . 9 ⊢ (ℤRHom‘ℝ_fld) = ( I ↾ ℤ)
11		f1eq1 6759	. . . . . . . . 9 ⊢ ((ℤRHom‘ℝ_fld) = ( I ↾ ℤ) → ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ ↔ ( I ↾ ℤ):ℤ–1-1→ℝ))
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ ↔ ( I ↾ ℤ):ℤ–1-1→ℝ)
13	9, 12	mpbir 234	. . . . . . 7 ⊢ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ
14		rebase 21716	. . . . . . . 8 ⊢ ℝ = (Base‘ℝ_fld)
15		eqid 2765	. . . . . . . 8 ⊢ (ℤRHom‘ℝ_fld) = (ℤRHom‘ℝ_fld)
16		re0g 21722	. . . . . . . 8 ⊢ 0 = (0_g‘ℝ_fld)
17	14, 15, 16	zrhchr 34281	. . . . . . 7 ⊢ (ℝ_fld ∈ Ring → ((chr‘ℝ_fld) = 0 ↔ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ))
18	13, 17	mpbiri 261	. . . . . 6 ⊢ (ℝ_fld ∈ Ring → (chr‘ℝ_fld) = 0)
19	2, 3, 18	mp2b 10	. . . . 5 ⊢ (chr‘ℝ_fld) = 0
20		eqid 2765	. . . . . 6 ⊢ (/_r‘ℝ_fld) = (/_r‘ℝ_fld)
21	14, 20, 15	qqhvval 34290	. . . . 5 ⊢ (((ℝ_fld ∈ DivRing ∧ (chr‘ℝ_fld) = 0) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘ℝ_fld)‘𝑞) = (((ℤRHom‘ℝ_fld)‘(numer‘𝑞))(/_r‘ℝ_fld)((ℤRHom‘ℝ_fld)‘(denom‘𝑞))))
22	2, 19, 21	mpanl12 714	. . . 4 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = (((ℤRHom‘ℝ_fld)‘(numer‘𝑞))(/_r‘ℝ_fld)((ℤRHom‘ℝ_fld)‘(denom‘𝑞))))
23		f1f 6764	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ → (ℤRHom‘ℝ_fld):ℤ⟶ℝ)
24	13, 23	ax-mp 5	. . . . . . 7 ⊢ (ℤRHom‘ℝ_fld):ℤ⟶ℝ
25	24	a1i 11	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (ℤRHom‘ℝ_fld):ℤ⟶ℝ)
26		qnumcl 16789	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (numer‘𝑞) ∈ ℤ)
27	25, 26	ffvelcdmd 7070	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) ∈ ℝ)
28		qdencl 16790	. . . . . . 7 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℕ)
29	28	nnzd 12608	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℤ)
30	25, 29	ffvelcdmd 7070	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ∈ ℝ)
31	29	anim1i 626	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
32	14, 15, 16	zrhf1ker 34280	. . . . . . . . . . . 12 ⊢ (ℝ_fld ∈ Ring → ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ ↔ (^◡(ℤRHom‘ℝ_fld) “ {0}) = {0}))
33	2, 3, 32	mp2b 10	. . . . . . . . . . 11 ⊢ ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ ↔ (^◡(ℤRHom‘ℝ_fld) “ {0}) = {0})
34	13, 33	mpbi 233	. . . . . . . . . 10 ⊢ (^◡(ℤRHom‘ℝ_fld) “ {0}) = {0}
35	34	eleq2i 2857	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ (^◡(ℤRHom‘ℝ_fld) “ {0}) ↔ (denom‘𝑞) ∈ {0})
36		ffn 6695	. . . . . . . . . 10 ⊢ ((ℤRHom‘ℝ_fld):ℤ⟶ℝ → (ℤRHom‘ℝ_fld) Fn ℤ)
37		fniniseg 7045	. . . . . . . . . 10 ⊢ ((ℤRHom‘ℝ_fld) Fn ℤ → ((denom‘𝑞) ∈ (^◡(ℤRHom‘ℝ_fld) “ {0}) ↔ ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0)))
38	24, 36, 37	mp2b 10	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ (^◡(ℤRHom‘ℝ_fld) “ {0}) ↔ ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
39		fvex 6884	. . . . . . . . . 10 ⊢ (denom‘𝑞) ∈ V
40	39	elsn 4600	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ {0} ↔ (denom‘𝑞) = 0)
41	35, 38, 40	3bitr3ri 305	. . . . . . . 8 ⊢ ((denom‘𝑞) = 0 ↔ ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
42	31, 41	sylibr 237	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → (denom‘𝑞) = 0)
43	28	nnne0d 12277	. . . . . . . . 9 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ≠ 0)
44	43	adantr 485	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → (denom‘𝑞) ≠ 0)
45	44	neneqd 2965	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ¬ (denom‘𝑞) = 0)
46	42, 45	pm2.65da 828	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ¬ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0)
47	46	neqned 2967	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ≠ 0)
48		redvr 21727	. . . . 5 ⊢ ((((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) ∈ ℝ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ∈ ℝ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ≠ 0) → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞))(/_r‘ℝ_fld)((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))))
49	27, 30, 47, 48	syl3anc 1394	. . . 4 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞))(/_r‘ℝ_fld)((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))))
50	10	fveq1i 6872	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (( I ↾ ℤ)‘(numer‘𝑞))
51		fvresi 7161	. . . . . . . 8 ⊢ ((numer‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(numer‘𝑞)) = (numer‘𝑞))
52	50, 51	eqtrid 2812	. . . . . . 7 ⊢ ((numer‘𝑞) ∈ ℤ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (numer‘𝑞))
53	26, 52	syl 18	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (numer‘𝑞))
54	10	fveq1i 6872	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (( I ↾ ℤ)‘(denom‘𝑞))
55		fvresi 7161	. . . . . . . 8 ⊢ ((denom‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(denom‘𝑞)) = (denom‘𝑞))
56	54, 55	eqtrid 2812	. . . . . . 7 ⊢ ((denom‘𝑞) ∈ ℤ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (denom‘𝑞))
57	29, 56	syl 18	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (denom‘𝑞))
58	53, 57	oveq12d 7418	. . . . 5 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = ((numer‘𝑞) / (denom‘𝑞)))
59		qeqnumdivden 16795	. . . . 5 ⊢ (𝑞 ∈ ℚ → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
60	58, 59	eqtr4d 2803	. . . 4 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = 𝑞)
61	22, 49, 60	3eqtrd 2804	. . 3 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = 𝑞)
62	61	mpteq2ia 5200	. 2 ⊢ (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)) = (𝑞 ∈ ℚ ↦ 𝑞)
63	14, 20, 15	qqhf 34293	. . . . . 6 ⊢ ((ℝ_fld ∈ DivRing ∧ (chr‘ℝ_fld) = 0) → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
64	2, 19, 63	mp2an 704	. . . . 5 ⊢ (ℚHom‘ℝ_fld):ℚ⟶ℝ
65	64	a1i 11	. . . 4 ⊢ (⊤ → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
66	65	feqmptd 6939	. . 3 ⊢ (⊤ → (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)))
67	66	mptru 1570	. 2 ⊢ (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞))
68		mptresid 6044	. 2 ⊢ ( I ↾ ℚ) = (𝑞 ∈ ℚ ↦ 𝑞)
69	62, 67, 68	3eqtr4i 2798	1 ⊢ (ℚHom‘ℝ_fld) = ( I ↾ ℚ)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 400 = wceq 1563 ⊤wtru 1564 ∈ wcel 2145 ≠ wne 2960 ⊆ wss 3907 {csn 4585 ↦ cmpt 5186 I cid 5546 ^◡ccnv 5651 ↾ cres 5654 “ cima 5655 Fn wfn 6520 ⟶wf 6521 –1-1→wf1 6522 –1-1-onto→wf1o 6524 ‘cfv 6525 (class class class)co 7400 ℝcr 11087 0cc0 11088 / cdiv 11859 ℤcz 12582 ℚcq 12963 numercnumer 16782 denomcdenom 16783 Ringcrg 20306 /_rcdvr 20473 SubRingcsubrg 20645 DivRingcdr 20804 ℂ_fldccnfld 21482 ℤRHomczrh 21609 chrcchr 21611 ℝ_fldcrefld 21714 ℚHomcqqh 34277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167 ax-mulf 11168

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-inf 9391 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-fz 13527 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-dvds 16301 df-gcd 16543 df-numer 16784 df-denom 16785 df-gz 16980 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-0g 17484 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-mhm 18831 df-grp 18993 df-minusg 18994 df-sbg 18995 df-mulg 19125 df-subg 19180 df-ghm 19275 df-od 19589 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-cring 20309 df-oppr 20410 df-dvdsr 20430 df-unit 20431 df-invr 20461 df-dvr 20474 df-rhm 20545 df-subrng 20622 df-subrg 20646 df-drng 20806 df-cnfld 21483 df-zring 21557 df-zrh 21613 df-chr 21615 df-refld 21715 df-qqh 34278

This theorem is referenced by: rrhre 34328

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