qqhre - Mathbox for Thierry Arnoux

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Theorem qqhre 31254

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 20744	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
2	1	simpri 488	. . . . 5 ⊢ ℝ_fld ∈ DivRing
3		drngring 19501	. . . . . 6 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
4		f1oi 6645	. . . . . . . . . 10 ⊢ ( I ↾ ℤ):ℤ–1-1-onto→ℤ
5		f1of1 6607	. . . . . . . . . 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 11980	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
8		f1ss 6573	. . . . . . . . 9 ⊢ ((( I ↾ ℤ):ℤ–1-1→ℤ ∧ ℤ ⊆ ℝ) → ( I ↾ ℤ):ℤ–1-1→ℝ)
9	6, 7, 8	mp2an 690	. . . . . . . 8 ⊢ ( I ↾ ℤ):ℤ–1-1→ℝ
10		zrhre 31253	. . . . . . . . 9 ⊢ (ℤRHom‘ℝ_fld) = ( I ↾ ℤ)
11		f1eq1 6563	. . . . . . . . 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 233	. . . . . . 7 ⊢ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ
14		rebase 20742	. . . . . . . 8 ⊢ ℝ = (Base‘ℝ_fld)
15		eqid 2819	. . . . . . . 8 ⊢ (ℤRHom‘ℝ_fld) = (ℤRHom‘ℝ_fld)
16		re0g 20748	. . . . . . . 8 ⊢ 0 = (0_g‘ℝ_fld)
17	14, 15, 16	zrhchr 31210	. . . . . . 7 ⊢ (ℝ_fld ∈ Ring → ((chr‘ℝ_fld) = 0 ↔ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ))
18	13, 17	mpbiri 260	. . . . . 6 ⊢ (ℝ_fld ∈ Ring → (chr‘ℝ_fld) = 0)
19	2, 3, 18	mp2b 10	. . . . 5 ⊢ (chr‘ℝ_fld) = 0
20		eqid 2819	. . . . . 6 ⊢ (/_r‘ℝ_fld) = (/_r‘ℝ_fld)
21	14, 20, 15	qqhvval 31217	. . . . 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 700	. . . 4 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = (((ℤRHom‘ℝ_fld)‘(numer‘𝑞))(/_r‘ℝ_fld)((ℤRHom‘ℝ_fld)‘(denom‘𝑞))))
23		f1f 6568	. . . . . . . 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 16072	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (numer‘𝑞) ∈ ℤ)
27	25, 26	ffvelrnd 6845	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) ∈ ℝ)
28		qdencl 16073	. . . . . . 7 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℕ)
29	28	nnzd 12078	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℤ)
30	25, 29	ffvelrnd 6845	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ∈ ℝ)
31	29	anim1i 616	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
32	14, 15, 16	zrhf1ker 31209	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (^◡(ℤRHom‘ℝ_fld) “ {0}) = {0}
35	34	eleq2i 2902	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ (^◡(ℤRHom‘ℝ_fld) “ {0}) ↔ (denom‘𝑞) ∈ {0})
36		ffn 6507	. . . . . . . . . 10 ⊢ ((ℤRHom‘ℝ_fld):ℤ⟶ℝ → (ℤRHom‘ℝ_fld) Fn ℤ)
37		fniniseg 6823	. . . . . . . . . 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 6676	. . . . . . . . . 10 ⊢ (denom‘𝑞) ∈ V
40	39	elsn 4574	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ {0} ↔ (denom‘𝑞) = 0)
41	35, 38, 40	3bitr3ri 304	. . . . . . . 8 ⊢ ((denom‘𝑞) = 0 ↔ ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
42	31, 41	sylibr 236	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → (denom‘𝑞) = 0)
43	28	nnne0d 11679	. . . . . . . . 9 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ≠ 0)
44	43	adantr 483	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → (denom‘𝑞) ≠ 0)
45	44	neneqd 3019	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ¬ (denom‘𝑞) = 0)
46	42, 45	pm2.65da 815	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ¬ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0)
47	46	neqned 3021	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ≠ 0)
48		redvr 20753	. . . . 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 1365	. . . 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 6664	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (( I ↾ ℤ)‘(numer‘𝑞))
51		fvresi 6928	. . . . . . . 8 ⊢ ((numer‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(numer‘𝑞)) = (numer‘𝑞))
52	50, 51	syl5eq 2866	. . . . . . 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 6664	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (( I ↾ ℤ)‘(denom‘𝑞))
55		fvresi 6928	. . . . . . . 8 ⊢ ((denom‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(denom‘𝑞)) = (denom‘𝑞))
56	54, 55	syl5eq 2866	. . . . . . 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 7166	. . . . 5 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = ((numer‘𝑞) / (denom‘𝑞)))
59		qeqnumdivden 16078	. . . . 5 ⊢ (𝑞 ∈ ℚ → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
60	58, 59	eqtr4d 2857	. . . 4 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = 𝑞)
61	22, 49, 60	3eqtrd 2858	. . 3 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = 𝑞)
62	61	mpteq2ia 5148	. 2 ⊢ (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)) = (𝑞 ∈ ℚ ↦ 𝑞)
63	14, 20, 15	qqhf 31220	. . . . . 6 ⊢ ((ℝ_fld ∈ DivRing ∧ (chr‘ℝ_fld) = 0) → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
64	2, 19, 63	mp2an 690	. . . . 5 ⊢ (ℚHom‘ℝ_fld):ℚ⟶ℝ
65	64	a1i 11	. . . 4 ⊢ (⊤ → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
66	65	feqmptd 6726	. . 3 ⊢ (⊤ → (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)))
67	66	mptru 1537	. 2 ⊢ (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞))
68		mptresid 5911	. 2 ⊢ ( I ↾ ℚ) = (𝑞 ∈ ℚ ↦ 𝑞)
69	62, 67, 68	3eqtr4i 2852	1 ⊢ (ℚHom‘ℝ_fld) = ( I ↾ ℚ)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1530 ⊤wtru 1531 ∈ wcel 2107 ≠ wne 3014 ⊆ wss 3934 {csn 4559 ↦ cmpt 5137 I cid 5452 ^◡ccnv 5547 ↾ cres 5550 “ cima 5551 Fn wfn 6343 ⟶wf 6344 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 0cc0 10529 / cdiv 11289 ℤcz 11973 ℚcq 12340 numercnumer 16065 denomcdenom 16066 Ringcrg 19289 /_rcdvr 19424 DivRingcdr 19494 SubRingcsubrg 19523 ℂ_fldccnfld 20537 ℤRHomczrh 20639 chrcchr 20641 ℝ_fldcrefld 20740 ℚHomcqqh 31206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1904 ax-6 1963 ax-7 2008 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2153 ax-12 2169 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1082 df-3an 1083 df-tru 1533 df-ex 1774 df-nf 1778 df-sb 2063 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-inf 8899 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-fz 12885 df-fl 13154 df-mod 13230 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15836 df-numer 16067 df-denom 16068 df-gz 16258 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-od 18648 df-cmn 18900 df-mgp 19232 df-ur 19244 df-ring 19291 df-cring 19292 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-dvr 19425 df-rnghom 19459 df-drng 19496 df-subrg 19525 df-cnfld 20538 df-zring 20610 df-zrh 20643 df-chr 20645 df-refld 20741 df-qqh 31207

This theorem is referenced by: rrhre 31255

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