cygznlem2a - Metamath Proof Explorer

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Theorem cygznlem2a 21520

Description: Lemma for cygzn 21523. (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 6847	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ V)
3		cygzn.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp)
4		cyggrp 19817	. . . . . . 7 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝐺 ∈ Grp)
7		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
8		cygzn.e	. . . . . . . 8 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
9	8	ssrab3 4032	. . . . . . 7 ⊢ 𝐸 ⊆ 𝐵
10		cygzn.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
11	9, 10	sselid 3929	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑋 ∈ 𝐵)
13		cygzn.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
14		cygzn.m	. . . . . 6 ⊢ · = (._g‘𝐺)
15	13, 14	mulgcl 19019	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
16	6, 7, 12, 15	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝑚 · 𝑋) ∈ 𝐵)
17		fveq2 6832	. . . 4 ⊢ (𝑚 = 𝑘 → (𝐿‘𝑚) = (𝐿‘𝑘))
18		oveq1 7363	. . . 4 ⊢ (𝑚 = 𝑘 → (𝑚 · 𝑋) = (𝑘 · 𝑋))
19		cygzn.n	. . . . . . . 8 ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
20		cygzn.y	. . . . . . . 8 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
21		cygzn.l	. . . . . . . 8 ⊢ 𝐿 = (ℤRHom‘𝑌)
22	13, 19, 20, 14, 21, 8, 3, 10	cygznlem1 21519	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) ↔ (𝑚 · 𝑋) = (𝑘 · 𝑋)))
23	22	biimpd 229	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋)))
24	23	exp32 420	. . . . 5 ⊢ (𝜑 → (𝑚 ∈ ℤ → (𝑘 ∈ ℤ → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋)))))
25	24	3imp2 1350	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ (𝐿‘𝑚) = (𝐿‘𝑘))) → (𝑚 · 𝑋) = (𝑘 · 𝑋))
26	1, 2, 16, 17, 18, 25	fliftfund 7257	. . 3 ⊢ (𝜑 → Fun 𝐹)
27	1, 2, 16	fliftf 7259	. . 3 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵))
28	26, 27	mpbid 232	. 2 ⊢ (𝜑 → 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵)
29		hashcl 14277	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
30	29	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ₀)
31		0nn0 12414	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
32	31	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ₀)
33	30, 32	ifclda 4513	. . . . . . . . 9 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ₀)
34	19, 33	eqeltrid 2838	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
35		eqid 2734	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
36	20, 35, 21	znzrhfo 21500	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑌))
37	34, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌))
38		fof 6744	. . . . . . 7 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
39	37, 38	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
40	39	feqmptd 6900	. . . . 5 ⊢ (𝜑 → 𝐿 = (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)))
41	40	rneqd 5885	. . . 4 ⊢ (𝜑 → ran 𝐿 = ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)))
42		forn 6747	. . . . 5 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → ran 𝐿 = (Base‘𝑌))
43	37, 42	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐿 = (Base‘𝑌))
44	41, 43	eqtr3d 2771	. . 3 ⊢ (𝜑 → ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)) = (Base‘𝑌))
45	44	feq2d 6644	. 2 ⊢ (𝜑 → (𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵 ↔ 𝐹:(Base‘𝑌)⟶𝐵))
46	28, 45	mpbid 232	1 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {crab 3397 Vcvv 3438 ifcif 4477 ⟨cop 4584 ↦ cmpt 5177 ran crn 5623 Fun wfun 6484 ⟶wf 6486 –onto→wfo 6488 ‘cfv 6490 (class class class)co 7356 Fincfn 8881 0cc0 11024 ℕ₀cn0 12399 ℤcz 12486 ♯chash 14251 Basecbs 17134 Grpcgrp 18861 ._gcmg 18995 CycGrpccyg 19804 ℤRHomczrh 21452 ℤ/nℤczn 21455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103 ax-mulf 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-omul 8400 df-er 8633 df-ec 8635 df-qs 8639 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-acn 9852 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-rp 12904 df-fz 13422 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-dvds 16178 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-0g 17359 df-imas 17427 df-qus 17428 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-subg 19051 df-nsg 19052 df-eqg 19053 df-ghm 19140 df-od 19455 df-cmn 19709 df-abl 19710 df-cyg 19805 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20271 df-dvdsr 20291 df-rhm 20406 df-subrng 20477 df-subrg 20501 df-lmod 20811 df-lss 20881 df-lsp 20921 df-sra 21123 df-rgmod 21124 df-lidl 21161 df-rsp 21162 df-2idl 21203 df-cnfld 21308 df-zring 21400 df-zrh 21456 df-zn 21459

This theorem is referenced by: cygznlem2 21521 cygznlem3 21522

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