cygznlem2a - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cygznlem2a		Structured version Visualization version GIF version

Theorem cygznlem2a 19916

Description: Lemma for cygzn 19919. (Contributed by Mario Carneiro, 23-Dec-2016.)

**Hypotheses**
Ref	Expression
cygzn.b	⊢ 𝐵 = (Base‘𝐺)
cygzn.n	⊢ 𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0)
cygzn.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)
cygzn.m	⊢ · = (._g‘𝐺)
cygzn.l	⊢ 𝐿 = (ℤRHom‘𝑌)
cygzn.e	⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
cygzn.g	⊢ (𝜑 → 𝐺 ∈ CycGrp)
cygzn.x	⊢ (𝜑 → 𝑋 ∈ 𝐸)
cygzn.f	⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿‘𝑚), (𝑚 · 𝑋)⟩)

**Assertion**
Ref	Expression
cygznlem2a	⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)

Distinct variable groups: 𝑚,𝑛,𝑥,𝐵 𝑚,𝐺,𝑛,𝑥 · ,𝑚,𝑛,𝑥 𝑚,𝑌,𝑛,𝑥 𝑚,𝐿,𝑛,𝑥 𝑥,𝑁 𝜑,𝑚 𝑛,𝐹,𝑥 𝑚,𝑋,𝑛,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑛) 𝐸(𝑥,𝑚,𝑛) 𝐹(𝑚) 𝑁(𝑚,𝑛)

Proof of Theorem cygznlem2a Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cygzn.f	. . . 4 ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿‘𝑚), (𝑚 · 𝑋)⟩)
2		fvexd 6203	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ V)
3		cygzn.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp)
4		cyggrp 18291	. . . . . . 7 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝐺 ∈ Grp)
7		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
8		cygzn.e	. . . . . . . 8 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
9		ssrab2 3687	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ⊆ 𝐵
10	8, 9	eqsstri 3635	. . . . . . 7 ⊢ 𝐸 ⊆ 𝐵
11		cygzn.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
12	10, 11	sseldi 3601	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
13	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑋 ∈ 𝐵)
14		cygzn.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
15		cygzn.m	. . . . . 6 ⊢ · = (._g‘𝐺)
16	14, 15	mulgcl 17559	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
17	6, 7, 13, 16	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝑚 · 𝑋) ∈ 𝐵)
18		fveq2 6191	. . . 4 ⊢ (𝑚 = 𝑘 → (𝐿‘𝑚) = (𝐿‘𝑘))
19		oveq1 6657	. . . 4 ⊢ (𝑚 = 𝑘 → (𝑚 · 𝑋) = (𝑘 · 𝑋))
20		cygzn.n	. . . . . . . 8 ⊢ 𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0)
21		cygzn.y	. . . . . . . 8 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
22		cygzn.l	. . . . . . . 8 ⊢ 𝐿 = (ℤRHom‘𝑌)
23	14, 20, 21, 15, 22, 8, 3, 11	cygznlem1 19915	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) ↔ (𝑚 · 𝑋) = (𝑘 · 𝑋)))
24	23	biimpd 219	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋)))
25	24	exp32 631	. . . . 5 ⊢ (𝜑 → (𝑚 ∈ ℤ → (𝑘 ∈ ℤ → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋)))))
26	25	3imp2 1282	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ (𝐿‘𝑚) = (𝐿‘𝑘))) → (𝑚 · 𝑋) = (𝑘 · 𝑋))
27	1, 2, 17, 18, 19, 26	fliftfund 6563	. . 3 ⊢ (𝜑 → Fun 𝐹)
28	1, 2, 17	fliftf 6565	. . 3 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵))
29	27, 28	mpbid 222	. 2 ⊢ (𝜑 → 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵)
30		hashcl 13147	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ₀)
31	30	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (#‘𝐵) ∈ ℕ₀)
32		0nn0 11307	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
33	32	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ₀)
34	31, 33	ifclda 4120	. . . . . . . . 9 ⊢ (𝜑 → if(𝐵 ∈ Fin, (#‘𝐵), 0) ∈ ℕ₀)
35	20, 34	syl5eqel 2705	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
36		eqid 2622	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
37	21, 36, 22	znzrhfo 19896	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑌))
38	35, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌))
39		fof 6115	. . . . . . 7 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
40	38, 39	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
41	40	feqmptd 6249	. . . . 5 ⊢ (𝜑 → 𝐿 = (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)))
42	41	rneqd 5353	. . . 4 ⊢ (𝜑 → ran 𝐿 = ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)))
43		forn 6118	. . . . 5 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → ran 𝐿 = (Base‘𝑌))
44	38, 43	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐿 = (Base‘𝑌))
45	42, 44	eqtr3d 2658	. . 3 ⊢ (𝜑 → ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)) = (Base‘𝑌))
46	45	feq2d 6031	. 2 ⊢ (𝜑 → (𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵 ↔ 𝐹:(Base‘𝑌)⟶𝐵))
47	29, 46	mpbid 222	1 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ifcif 4086 ⟨cop 4183 ↦ cmpt 4729 ran crn 5115 Fun wfun 5882 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 0cc0 9936 ℕ₀cn0 11292 ℤcz 11377 #chash 13117 Basecbs 15857 Grpcgrp 17422 ._gcmg 17540 CycGrpccyg 18279 ℤRHomczrh 19848 ℤ/nℤczn 19851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-imas 16168 df-qus 16169 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-od 17948 df-cmn 18195 df-abl 18196 df-cyg 18280 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-zn 19855

This theorem is referenced by: cygznlem2 19917 cygznlem3 19918

W3C validator