cygznlem2a - Metamath Proof Explorer

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Theorem cygznlem2a 21885

Description: Lemma for cygzn 21888. (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 6908	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ V)
3		cygzn.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp)
4		cyggrp 20116	. . . . . . 7 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
5	3, 4	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
6	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝐺 ∈ Grp)
7		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
8		cygzn.e	. . . . . . . 8 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
9	8	ssrab3 4009	. . . . . . 7 ⊢ 𝐸 ⊆ 𝐵
10		cygzn.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
11	9, 10	sselid 3910	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → 𝑋 ∈ 𝐵)
13		cygzn.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
14		cygzn.m	. . . . . 6 ⊢ · = (._g‘𝐺)
15	13, 14	mulgcl 19313	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑚 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑚 · 𝑋) ∈ 𝐵)
16	6, 7, 12, 15	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (𝑚 · 𝑋) ∈ 𝐵)
17		fveq2 6893	. . . 4 ⊢ (𝑚 = 𝑘 → (𝐿‘𝑚) = (𝐿‘𝑘))
18		oveq1 7434	. . . 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 21884	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) ↔ (𝑚 · 𝑋) = (𝑘 · 𝑋)))
23	22	biimpd 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋)))
24	23	exp32 426	. . . . 5 ⊢ (𝜑 → (𝑚 ∈ ℤ → (𝑘 ∈ ℤ → ((𝐿‘𝑚) = (𝐿‘𝑘) → (𝑚 · 𝑋) = (𝑘 · 𝑋)))))
25	24	3imp2 1368	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ (𝐿‘𝑚) = (𝐿‘𝑘))) → (𝑚 · 𝑋) = (𝑘 · 𝑋))
26	1, 2, 16, 17, 18, 25	fliftfund 7328	. . 3 ⊢ (𝜑 → Fun 𝐹)
27	1, 2, 16	fliftf 7330	. . 3 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵))
28	26, 27	mpbid 235	. 2 ⊢ (𝜑 → 𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵)
29		hashcl 14510	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
30	29	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ₀)
31		0nn0 12631	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
32	31	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ₀)
33	30, 32	ifclda 4520	. . . . . . . . 9 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ₀)
34	19, 33	eqeltrid 2867	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
35		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
36	20, 35, 21	znzrhfo 21865	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑌))
37	34, 36	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌))
38		fof 6804	. . . . . . 7 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
39	37, 38	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑌))
40	39	feqmptd 6961	. . . . 5 ⊢ (𝜑 → 𝐿 = (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)))
41	40	rneqd 5924	. . . 4 ⊢ (𝜑 → ran 𝐿 = ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)))
42		forn 6807	. . . . 5 ⊢ (𝐿:ℤ–onto→(Base‘𝑌) → ran 𝐿 = (Base‘𝑌))
43	37, 42	syl 18	. . . 4 ⊢ (𝜑 → ran 𝐿 = (Base‘𝑌))
44	41, 43	eqtr3d 2800	. . 3 ⊢ (𝜑 → ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚)) = (Base‘𝑌))
45	44	feq2d 6701	. 2 ⊢ (𝜑 → (𝐹:ran (𝑚 ∈ ℤ ↦ (𝐿‘𝑚))⟶𝐵 ↔ 𝐹:(Base‘𝑌)⟶𝐵))
46	28, 45	mpbid 235	1 ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2146 {crab 3415 Vcvv 3453 ifcif 4484 ⟨cop 4837 ↦ cmpt 5188 ran crn 5655 Fun wfun 6540 ⟶wf 6542 –onto→wfo 6544 ‘cfv 6546 (class class class)co 7427 Fincfn 8984 0cc0 11210 ℕ₀cn0 12616 ℤcz 12703 ♯chash 14484 Basecbs 17399 Grpcgrp 19156 ._gcmg 19289 CycGrpccyg 20103 ℤRHomczrh 21817 ℤ/nℤczn 21820

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-inf2 9652 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288 ax-addf 11289 ax-mulf 11290

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-tpos 8254 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8728 df-ec 8730 df-qs 8734 df-map 8860 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9444 df-inf 9445 df-oi 9514 df-card 10030 df-acn 10033 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-z 12704 df-dec 12825 df-uz 12976 df-rp 13131 df-fz 13650 df-fl 13942 df-mod 14020 df-seq 14155 df-exp 14215 df-hash 14485 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-dvds 16435 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-starv 17455 df-sca 17456 df-vsca 17457 df-ip 17458 df-tset 17459 df-ple 17460 df-ds 17462 df-unif 17463 df-0g 17624 df-imas 17692 df-qus 17693 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-mhm 18990 df-grp 19159 df-minusg 19160 df-sbg 19161 df-mulg 19290 df-subg 19345 df-nsg 19346 df-eqg 19347 df-ghm 19440 df-od 19754 df-cmn 20008 df-abl 20009 df-cyg 20104 df-mgp 20373 df-rng 20387 df-ur 20420 df-ring 20473 df-cring 20474 df-oppr 20579 df-dvdsr 20599 df-rhm 20714 df-subrng 20810 df-subrg 20834 df-lmod 21149 df-lss 21219 df-lsp 21259 df-sra 21460 df-rgmod 21461 df-lidl 21498 df-rsp 21499 df-2idl 21555 df-cnfld 21691 df-zring 21765 df-zrh 21821 df-zn 21824

This theorem is used by: cygznlem2 21886 cygznlem3 21887

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