qqhre - Mathbox for Thierry Arnoux

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Theorem qqhre 31870

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 20725	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
2	1	simpri 485	. . . . 5 ⊢ ℝ_fld ∈ DivRing
3		drngring 19913	. . . . . 6 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
4		f1oi 6737	. . . . . . . . . 10 ⊢ ( I ↾ ℤ):ℤ–1-1-onto→ℤ
5		f1of1 6699	. . . . . . . . . 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 12256	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
8		f1ss 6660	. . . . . . . . 9 ⊢ ((( I ↾ ℤ):ℤ–1-1→ℤ ∧ ℤ ⊆ ℝ) → ( I ↾ ℤ):ℤ–1-1→ℝ)
9	6, 7, 8	mp2an 688	. . . . . . . 8 ⊢ ( I ↾ ℤ):ℤ–1-1→ℝ
10		zrhre 31869	. . . . . . . . 9 ⊢ (ℤRHom‘ℝ_fld) = ( I ↾ ℤ)
11		f1eq1 6649	. . . . . . . . 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 230	. . . . . . 7 ⊢ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ
14		rebase 20723	. . . . . . . 8 ⊢ ℝ = (Base‘ℝ_fld)
15		eqid 2738	. . . . . . . 8 ⊢ (ℤRHom‘ℝ_fld) = (ℤRHom‘ℝ_fld)
16		re0g 20729	. . . . . . . 8 ⊢ 0 = (0_g‘ℝ_fld)
17	14, 15, 16	zrhchr 31826	. . . . . . 7 ⊢ (ℝ_fld ∈ Ring → ((chr‘ℝ_fld) = 0 ↔ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ))
18	13, 17	mpbiri 257	. . . . . 6 ⊢ (ℝ_fld ∈ Ring → (chr‘ℝ_fld) = 0)
19	2, 3, 18	mp2b 10	. . . . 5 ⊢ (chr‘ℝ_fld) = 0
20		eqid 2738	. . . . . 6 ⊢ (/_r‘ℝ_fld) = (/_r‘ℝ_fld)
21	14, 20, 15	qqhvval 31833	. . . . 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 698	. . . 4 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = (((ℤRHom‘ℝ_fld)‘(numer‘𝑞))(/_r‘ℝ_fld)((ℤRHom‘ℝ_fld)‘(denom‘𝑞))))
23		f1f 6654	. . . . . . . 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 16372	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (numer‘𝑞) ∈ ℤ)
27	25, 26	ffvelrnd 6944	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) ∈ ℝ)
28		qdencl 16373	. . . . . . 7 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℕ)
29	28	nnzd 12354	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℤ)
30	25, 29	ffvelrnd 6944	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ∈ ℝ)
31	29	anim1i 614	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
32	14, 15, 16	zrhf1ker 31825	. . . . . . . . . . . 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 229	. . . . . . . . . 10 ⊢ (^◡(ℤRHom‘ℝ_fld) “ {0}) = {0}
35	34	eleq2i 2830	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ (^◡(ℤRHom‘ℝ_fld) “ {0}) ↔ (denom‘𝑞) ∈ {0})
36		ffn 6584	. . . . . . . . . 10 ⊢ ((ℤRHom‘ℝ_fld):ℤ⟶ℝ → (ℤRHom‘ℝ_fld) Fn ℤ)
37		fniniseg 6919	. . . . . . . . . 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 6769	. . . . . . . . . 10 ⊢ (denom‘𝑞) ∈ V
40	39	elsn 4573	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ {0} ↔ (denom‘𝑞) = 0)
41	35, 38, 40	3bitr3ri 301	. . . . . . . 8 ⊢ ((denom‘𝑞) = 0 ↔ ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
42	31, 41	sylibr 233	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → (denom‘𝑞) = 0)
43	28	nnne0d 11953	. . . . . . . . 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 2947	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ¬ (denom‘𝑞) = 0)
46	42, 45	pm2.65da 813	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ¬ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0)
47	46	neqned 2949	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ≠ 0)
48		redvr 20734	. . . . 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 1369	. . . 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 6757	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (( I ↾ ℤ)‘(numer‘𝑞))
51		fvresi 7027	. . . . . . . 8 ⊢ ((numer‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(numer‘𝑞)) = (numer‘𝑞))
52	50, 51	syl5eq 2791	. . . . . . 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 6757	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (( I ↾ ℤ)‘(denom‘𝑞))
55		fvresi 7027	. . . . . . . 8 ⊢ ((denom‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(denom‘𝑞)) = (denom‘𝑞))
56	54, 55	syl5eq 2791	. . . . . . 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 7273	. . . . 5 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = ((numer‘𝑞) / (denom‘𝑞)))
59		qeqnumdivden 16378	. . . . 5 ⊢ (𝑞 ∈ ℚ → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
60	58, 59	eqtr4d 2781	. . . 4 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = 𝑞)
61	22, 49, 60	3eqtrd 2782	. . 3 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = 𝑞)
62	61	mpteq2ia 5173	. 2 ⊢ (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)) = (𝑞 ∈ ℚ ↦ 𝑞)
63	14, 20, 15	qqhf 31836	. . . . . 6 ⊢ ((ℝ_fld ∈ DivRing ∧ (chr‘ℝ_fld) = 0) → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
64	2, 19, 63	mp2an 688	. . . . 5 ⊢ (ℚHom‘ℝ_fld):ℚ⟶ℝ
65	64	a1i 11	. . . 4 ⊢ (⊤ → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
66	65	feqmptd 6819	. . 3 ⊢ (⊤ → (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)))
67	66	mptru 1546	. 2 ⊢ (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞))
68		mptresid 5947	. 2 ⊢ ( I ↾ ℚ) = (𝑞 ∈ ℚ ↦ 𝑞)
69	62, 67, 68	3eqtr4i 2776	1 ⊢ (ℚHom‘ℝ_fld) = ( I ↾ ℚ)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ≠ wne 2942 ⊆ wss 3883 {csn 4558 ↦ cmpt 5153 I cid 5479 ^◡ccnv 5579 ↾ cres 5582 “ cima 5583 Fn wfn 6413 ⟶wf 6414 –1-1→wf1 6415 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 / cdiv 11562 ℤcz 12249 ℚcq 12617 numercnumer 16365 denomcdenom 16366 Ringcrg 19698 /_rcdvr 19839 DivRingcdr 19906 SubRingcsubrg 19935 ℂ_fldccnfld 20510 ℤRHomczrh 20613 chrcchr 20615 ℝ_fldcrefld 20721 ℚHomcqqh 31822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-fz 13169 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-numer 16367 df-denom 16368 df-gz 16559 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-od 19051 df-cmn 19303 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-chr 20619 df-refld 20722 df-qqh 31823

This theorem is referenced by: rrhre 31871

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