qqhre - Mathbox for Thierry Arnoux

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Theorem qqhre 34021

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 21626	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
2	1	simpri 485	. . . . 5 ⊢ ℝ_fld ∈ DivRing
3		drngring 20736	. . . . . 6 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
4		f1oi 6886	. . . . . . . . . 10 ⊢ ( I ↾ ℤ):ℤ–1-1-onto→ℤ
5		f1of1 6847	. . . . . . . . . 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 12620	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
8		f1ss 6809	. . . . . . . . 9 ⊢ ((( I ↾ ℤ):ℤ–1-1→ℤ ∧ ℤ ⊆ ℝ) → ( I ↾ ℤ):ℤ–1-1→ℝ)
9	6, 7, 8	mp2an 692	. . . . . . . 8 ⊢ ( I ↾ ℤ):ℤ–1-1→ℝ
10		zrhre 34020	. . . . . . . . 9 ⊢ (ℤRHom‘ℝ_fld) = ( I ↾ ℤ)
11		f1eq1 6799	. . . . . . . . 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 231	. . . . . . 7 ⊢ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ
14		rebase 21624	. . . . . . . 8 ⊢ ℝ = (Base‘ℝ_fld)
15		eqid 2737	. . . . . . . 8 ⊢ (ℤRHom‘ℝ_fld) = (ℤRHom‘ℝ_fld)
16		re0g 21630	. . . . . . . 8 ⊢ 0 = (0_g‘ℝ_fld)
17	14, 15, 16	zrhchr 33975	. . . . . . 7 ⊢ (ℝ_fld ∈ Ring → ((chr‘ℝ_fld) = 0 ↔ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ))
18	13, 17	mpbiri 258	. . . . . 6 ⊢ (ℝ_fld ∈ Ring → (chr‘ℝ_fld) = 0)
19	2, 3, 18	mp2b 10	. . . . 5 ⊢ (chr‘ℝ_fld) = 0
20		eqid 2737	. . . . . 6 ⊢ (/_r‘ℝ_fld) = (/_r‘ℝ_fld)
21	14, 20, 15	qqhvval 33984	. . . . 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 702	. . . 4 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = (((ℤRHom‘ℝ_fld)‘(numer‘𝑞))(/_r‘ℝ_fld)((ℤRHom‘ℝ_fld)‘(denom‘𝑞))))
23		f1f 6804	. . . . . . . 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 16777	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (numer‘𝑞) ∈ ℤ)
27	25, 26	ffvelcdmd 7105	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) ∈ ℝ)
28		qdencl 16778	. . . . . . 7 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℕ)
29	28	nnzd 12640	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℤ)
30	25, 29	ffvelcdmd 7105	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ∈ ℝ)
31	29	anim1i 615	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
32	14, 15, 16	zrhf1ker 33974	. . . . . . . . . . . 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 230	. . . . . . . . . 10 ⊢ (^◡(ℤRHom‘ℝ_fld) “ {0}) = {0}
35	34	eleq2i 2833	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ (^◡(ℤRHom‘ℝ_fld) “ {0}) ↔ (denom‘𝑞) ∈ {0})
36		ffn 6736	. . . . . . . . . 10 ⊢ ((ℤRHom‘ℝ_fld):ℤ⟶ℝ → (ℤRHom‘ℝ_fld) Fn ℤ)
37		fniniseg 7080	. . . . . . . . . 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 6919	. . . . . . . . . 10 ⊢ (denom‘𝑞) ∈ V
40	39	elsn 4641	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ {0} ↔ (denom‘𝑞) = 0)
41	35, 38, 40	3bitr3ri 302	. . . . . . . 8 ⊢ ((denom‘𝑞) = 0 ↔ ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
42	31, 41	sylibr 234	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → (denom‘𝑞) = 0)
43	28	nnne0d 12316	. . . . . . . . 9 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ≠ 0)
44	43	adantr 480	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → (denom‘𝑞) ≠ 0)
45	44	neneqd 2945	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ¬ (denom‘𝑞) = 0)
46	42, 45	pm2.65da 817	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ¬ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0)
47	46	neqned 2947	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ≠ 0)
48		redvr 21635	. . . . 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 1373	. . . 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 6907	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (( I ↾ ℤ)‘(numer‘𝑞))
51		fvresi 7193	. . . . . . . 8 ⊢ ((numer‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(numer‘𝑞)) = (numer‘𝑞))
52	50, 51	eqtrid 2789	. . . . . . 7 ⊢ ((numer‘𝑞) ∈ ℤ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (numer‘𝑞))
53	26, 52	syl 17	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (numer‘𝑞))
54	10	fveq1i 6907	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (( I ↾ ℤ)‘(denom‘𝑞))
55		fvresi 7193	. . . . . . . 8 ⊢ ((denom‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(denom‘𝑞)) = (denom‘𝑞))
56	54, 55	eqtrid 2789	. . . . . . 7 ⊢ ((denom‘𝑞) ∈ ℤ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (denom‘𝑞))
57	29, 56	syl 17	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (denom‘𝑞))
58	53, 57	oveq12d 7449	. . . . 5 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = ((numer‘𝑞) / (denom‘𝑞)))
59		qeqnumdivden 16783	. . . . 5 ⊢ (𝑞 ∈ ℚ → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
60	58, 59	eqtr4d 2780	. . . 4 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = 𝑞)
61	22, 49, 60	3eqtrd 2781	. . 3 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = 𝑞)
62	61	mpteq2ia 5245	. 2 ⊢ (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)) = (𝑞 ∈ ℚ ↦ 𝑞)
63	14, 20, 15	qqhf 33987	. . . . . 6 ⊢ ((ℝ_fld ∈ DivRing ∧ (chr‘ℝ_fld) = 0) → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
64	2, 19, 63	mp2an 692	. . . . 5 ⊢ (ℚHom‘ℝ_fld):ℚ⟶ℝ
65	64	a1i 11	. . . 4 ⊢ (⊤ → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
66	65	feqmptd 6977	. . 3 ⊢ (⊤ → (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)))
67	66	mptru 1547	. 2 ⊢ (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞))
68		mptresid 6069	. 2 ⊢ ( I ↾ ℚ) = (𝑞 ∈ ℚ ↦ 𝑞)
69	62, 67, 68	3eqtr4i 2775	1 ⊢ (ℚHom‘ℝ_fld) = ( I ↾ ℚ)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 {csn 4626 ↦ cmpt 5225 I cid 5577 ^◡ccnv 5684 ↾ cres 5687 “ cima 5688 Fn wfn 6556 ⟶wf 6557 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 / cdiv 11920 ℤcz 12613 ℚcq 12990 numercnumer 16770 denomcdenom 16771 Ringcrg 20230 /_rcdvr 20400 SubRingcsubrg 20569 DivRingcdr 20729 ℂ_fldccnfld 21364 ℤRHomczrh 21510 chrcchr 21512 ℝ_fldcrefld 21622 ℚHomcqqh 33971

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-fz 13548 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-numer 16772 df-denom 16773 df-gz 16968 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-od 19546 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-chr 21516 df-refld 21623 df-qqh 33972

This theorem is referenced by: rrhre 34022

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