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Theorem cygznlem2a 21477

Description: Lemma for cygzn 21480. (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 6873	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ V)
3		cygzn.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp)
4		cyggrp 19820	. . . . . . 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 4045	. . . . . . 7 ⊢ 𝐸 ⊆ 𝐵
10		cygzn.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
11	9, 10	sselid 3944	. . . . . 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 19023	. . . . 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 6858	. . . 4 ⊢ (𝑚 = 𝑘 → (𝐿‘𝑚) = (𝐿‘𝑘))
18		oveq1 7394	. . . 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 21476	. . . . . . 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 7288	. . 3 ⊢ (𝜑 → Fun 𝐹)
27	1, 2, 16	fliftf 7290	. . 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 14321	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
30	29	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ₀)
31		0nn0 12457	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
32	31	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ₀)
33	30, 32	ifclda 4524	. . . . . . . . 9 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ₀)
34	19, 33	eqeltrid 2832	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
35		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
36	20, 35, 21	znzrhfo 21457	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑌))
37	34, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌))
38		fof 6772	. . . . . . 7 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
39	37, 38	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
40	39	feqmptd 6929	. . . . 5 ⊢ (𝜑 → 𝐿 = (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)))
41	40	rneqd 5902	. . . 4 ⊢ (𝜑 → ran 𝐿 = ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)))
42		forn 6775	. . . . 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 2766	. . 3 ⊢ (𝜑 → ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)) = (Base‘𝑌))
45	44	feq2d 6672	. 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 1540 ∈ wcel 2109 {crab 3405 Vcvv 3447 ifcif 4488 ⟨cop 4595 ↦ cmpt 5188 ran crn 5639 Fun wfun 6505 ⟶wf 6507 –onto→wfo 6509 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 0cc0 11068 ℕ₀cn0 12442 ℤcz 12529 ♯chash 14295 Basecbs 17179 Grpcgrp 18865 ._gcmg 18999 CycGrpccyg 19807 ℤRHomczrh 21409 ℤ/nℤczn 21412

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-omul 8439 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-rp 12952 df-fz 13469 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-0g 17404 df-imas 17471 df-qus 17472 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19145 df-od 19458 df-cmn 19712 df-abl 19713 df-cyg 19808 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-lmod 20768 df-lss 20838 df-lsp 20878 df-sra 21080 df-rgmod 21081 df-lidl 21118 df-rsp 21119 df-2idl 21160 df-cnfld 21265 df-zring 21357 df-zrh 21413 df-zn 21416

This theorem is referenced by: cygznlem2 21478 cygznlem3 21479

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