qqhre - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > qqhre		Structured version Visualization version GIF version

Theorem qqhre 29392

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 19773	. . . . . . 7 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
2	1	simpri 477	. . . . . 6 ⊢ ℝ_fld ∈ DivRing
3		drngring 18577	. . . . . . 7 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
4		f1oi 6086	. . . . . . . . . . 11 ⊢ ( I ↾ ℤ):ℤ–1-1-onto→ℤ
5		f1of1 6049	. . . . . . . . . . 11 ⊢ (( I ↾ ℤ):ℤ–1-1-onto→ℤ → ( I ↾ ℤ):ℤ–1-1→ℤ)
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ↾ ℤ):ℤ–1-1→ℤ
7		zssre 11261	. . . . . . . . . 10 ⊢ ℤ ⊆ ℝ
8		f1ss 6019	. . . . . . . . . 10 ⊢ ((( I ↾ ℤ):ℤ–1-1→ℤ ∧ ℤ ⊆ ℝ) → ( I ↾ ℤ):ℤ–1-1→ℝ)
9	6, 7, 8	mp2an 704	. . . . . . . . 9 ⊢ ( I ↾ ℤ):ℤ–1-1→ℝ
10		zrhre 29391	. . . . . . . . . 10 ⊢ (ℤRHom‘ℝ_fld) = ( I ↾ ℤ)
11		f1eq1 6009	. . . . . . . . . 10 ⊢ ((ℤRHom‘ℝ_fld) = ( I ↾ ℤ) → ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ ↔ ( I ↾ ℤ):ℤ–1-1→ℝ))
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ ↔ ( I ↾ ℤ):ℤ–1-1→ℝ)
13	9, 12	mpbir 220	. . . . . . . 8 ⊢ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ
14		rebase 19771	. . . . . . . . 9 ⊢ ℝ = (Base‘ℝ_fld)
15		eqid 2610	. . . . . . . . 9 ⊢ (ℤRHom‘ℝ_fld) = (ℤRHom‘ℝ_fld)
16		re0g 19777	. . . . . . . . 9 ⊢ 0 = (0_g‘ℝ_fld)
17	14, 15, 16	zrhchr 29348	. . . . . . . 8 ⊢ (ℝ_fld ∈ Ring → ((chr‘ℝ_fld) = 0 ↔ (ℤRHom‘ℝ_fld):ℤ–1-1→ℝ))
18	13, 17	mpbiri 247	. . . . . . 7 ⊢ (ℝ_fld ∈ Ring → (chr‘ℝ_fld) = 0)
19	2, 3, 18	mp2b 10	. . . . . 6 ⊢ (chr‘ℝ_fld) = 0
20		eqid 2610	. . . . . . 7 ⊢ (/_r‘ℝ_fld) = (/_r‘ℝ_fld)
21	14, 20, 15	qqhf 29358	. . . . . 6 ⊢ ((ℝ_fld ∈ DivRing ∧ (chr‘ℝ_fld) = 0) → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
22	2, 19, 21	mp2an 704	. . . . 5 ⊢ (ℚHom‘ℝ_fld):ℚ⟶ℝ
23	22	a1i 11	. . . 4 ⊢ (⊤ → (ℚHom‘ℝ_fld):ℚ⟶ℝ)
24	23	feqmptd 6159	. . 3 ⊢ (⊤ → (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)))
25	24	trud 1484	. 2 ⊢ (ℚHom‘ℝ_fld) = (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞))
26	14, 20, 15	qqhvval 29355	. . . . 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))))
27	2, 19, 26	mpanl12 714	. . . 4 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = (((ℤRHom‘ℝ_fld)‘(numer‘𝑞))(/_r‘ℝ_fld)((ℤRHom‘ℝ_fld)‘(denom‘𝑞))))
28		f1f 6014	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ → (ℤRHom‘ℝ_fld):ℤ⟶ℝ)
29	13, 28	ax-mp 5	. . . . . . 7 ⊢ (ℤRHom‘ℝ_fld):ℤ⟶ℝ
30	29	a1i 11	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (ℤRHom‘ℝ_fld):ℤ⟶ℝ)
31		qnumcl 15286	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (numer‘𝑞) ∈ ℤ)
32	30, 31	ffvelrnd 6268	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) ∈ ℝ)
33		qdencl 15287	. . . . . . 7 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℕ)
34	33	nnzd 11357	. . . . . 6 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℤ)
35	30, 34	ffvelrnd 6268	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ∈ ℝ)
36	34	anim1i 590	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
37	14, 15, 16	zrhf1ker 29347	. . . . . . . . . . . 12 ⊢ (ℝ_fld ∈ Ring → ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ ↔ (^◡(ℤRHom‘ℝ_fld) “ {0}) = {0}))
38	2, 3, 37	mp2b 10	. . . . . . . . . . 11 ⊢ ((ℤRHom‘ℝ_fld):ℤ–1-1→ℝ ↔ (^◡(ℤRHom‘ℝ_fld) “ {0}) = {0})
39	13, 38	mpbi 219	. . . . . . . . . 10 ⊢ (^◡(ℤRHom‘ℝ_fld) “ {0}) = {0}
40	39	eleq2i 2680	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ (^◡(ℤRHom‘ℝ_fld) “ {0}) ↔ (denom‘𝑞) ∈ {0})
41		ffn 5958	. . . . . . . . . 10 ⊢ ((ℤRHom‘ℝ_fld):ℤ⟶ℝ → (ℤRHom‘ℝ_fld) Fn ℤ)
42		fniniseg 6246	. . . . . . . . . 10 ⊢ ((ℤRHom‘ℝ_fld) Fn ℤ → ((denom‘𝑞) ∈ (^◡(ℤRHom‘ℝ_fld) “ {0}) ↔ ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0)))
43	29, 41, 42	mp2b 10	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ (^◡(ℤRHom‘ℝ_fld) “ {0}) ↔ ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
44		fvex 6113	. . . . . . . . . 10 ⊢ (denom‘𝑞) ∈ V
45	44	elsn 4140	. . . . . . . . 9 ⊢ ((denom‘𝑞) ∈ {0} ↔ (denom‘𝑞) = 0)
46	40, 43, 45	3bitr3ri 290	. . . . . . . 8 ⊢ ((denom‘𝑞) = 0 ↔ ((denom‘𝑞) ∈ ℤ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0))
47	36, 46	sylibr 223	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → (denom‘𝑞) = 0)
48	33	nnne0d 10942	. . . . . . . . 9 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ≠ 0)
49	48	adantr 480	. . . . . . . 8 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → (denom‘𝑞) ≠ 0)
50	49	neneqd 2787	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0) → ¬ (denom‘𝑞) = 0)
51	47, 50	pm2.65da 598	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ¬ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = 0)
52	51	neqned 2789	. . . . 5 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) ≠ 0)
53		redvr 19782	. . . . 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))))
54	32, 35, 52, 53	syl3anc 1318	. . . 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))))
55	10	fveq1i 6104	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (( I ↾ ℤ)‘(numer‘𝑞))
56		fvresi 6344	. . . . . . . 8 ⊢ ((numer‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(numer‘𝑞)) = (numer‘𝑞))
57	55, 56	syl5eq 2656	. . . . . . 7 ⊢ ((numer‘𝑞) ∈ ℤ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (numer‘𝑞))
58	31, 57	syl 17	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) = (numer‘𝑞))
59	10	fveq1i 6104	. . . . . . . 8 ⊢ ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (( I ↾ ℤ)‘(denom‘𝑞))
60		fvresi 6344	. . . . . . . 8 ⊢ ((denom‘𝑞) ∈ ℤ → (( I ↾ ℤ)‘(denom‘𝑞)) = (denom‘𝑞))
61	59, 60	syl5eq 2656	. . . . . . 7 ⊢ ((denom‘𝑞) ∈ ℤ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (denom‘𝑞))
62	34, 61	syl 17	. . . . . 6 ⊢ (𝑞 ∈ ℚ → ((ℤRHom‘ℝ_fld)‘(denom‘𝑞)) = (denom‘𝑞))
63	58, 62	oveq12d 6567	. . . . 5 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = ((numer‘𝑞) / (denom‘𝑞)))
64		qeqnumdivden 15292	. . . . 5 ⊢ (𝑞 ∈ ℚ → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
65	63, 64	eqtr4d 2647	. . . 4 ⊢ (𝑞 ∈ ℚ → (((ℤRHom‘ℝ_fld)‘(numer‘𝑞)) / ((ℤRHom‘ℝ_fld)‘(denom‘𝑞))) = 𝑞)
66	27, 54, 65	3eqtrd 2648	. . 3 ⊢ (𝑞 ∈ ℚ → ((ℚHom‘ℝ_fld)‘𝑞) = 𝑞)
67	66	mpteq2ia 4668	. 2 ⊢ (𝑞 ∈ ℚ ↦ ((ℚHom‘ℝ_fld)‘𝑞)) = (𝑞 ∈ ℚ ↦ 𝑞)
68		mptresid 5375	. 2 ⊢ (𝑞 ∈ ℚ ↦ 𝑞) = ( I ↾ ℚ)
69	25, 67, 68	3eqtri 2636	1 ⊢ (ℚHom‘ℝ_fld) = ( I ↾ ℚ)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 {csn 4125 ↦ cmpt 4643 I cid 4948 ^◡ccnv 5037 ↾ cres 5040 “ cima 5041 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 / cdiv 10563 ℤcz 11254 ℚcq 11664 numercnumer 15279 denomcdenom 15280 Ringcrg 18370 /_rcdvr 18505 DivRingcdr 18570 SubRingcsubrg 18599 ℂ_fldccnfld 19567 ℤRHomczrh 19667 chrcchr 19669 ℝ_fldcrefld 19769 ℚHomcqqh 29344

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-numer 15281 df-denom 15282 df-gz 15472 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-od 17771 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-subrg 18601 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-chr 19673 df-refld 19770 df-qqh 29345

This theorem is referenced by: rrhre 29393

W3C validator