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Theorem cygznlem1 21778

Description: Lemma for cygzn 21782. (Contributed by Mario Carneiro, 21-Apr-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)

**Assertion**
Ref	Expression
cygznlem1	⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋)))

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

**Proof of Theorem cygznlem1**
Step	Hyp	Ref	Expression
1		cygzn.n	. . . . 5 ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
2		hashcl 14420	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
3	2	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ₀)
4		0nn0 12544	. . . . . . 7 ⊢ 0 ∈ ℕ₀
5	4	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ₀)
6	3, 5	ifclda 4521	. . . . 5 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ₀)
7	1, 6	eqeltrid 2866	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℕ₀)
9		simprl 783	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐾 ∈ ℤ)
10		simprr 785	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑀 ∈ ℤ)
11		cygzn.y	. . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
12		cygzn.l	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌)
13	11, 12	zndvds 21761	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀)))
14	8, 9, 10, 13	syl3anc 1398	. 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀)))
15		cygzn.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp)
16		cyggrp 20016	. . . . . . 7 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
17	15, 16	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
18		cygzn.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
19		cygzn.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
20		cygzn.m	. . . . . . 7 ⊢ · = (._g‘𝐺)
21		cygzn.e	. . . . . . 7 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
22		eqid 2762	. . . . . . 7 ⊢ (od‘𝐺) = (od‘𝐺)
23	19, 20, 21, 22	cyggenod2 20011	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0))
24	17, 18, 23	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0))
25	24, 1	eqtr4di 2815	. . . 4 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = 𝑁)
26	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((od‘𝐺)‘𝑋) = 𝑁)
27	26	breq1d 5117	. 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀)))
28	17	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐺 ∈ Grp)
29	19, 20, 21	iscyggen 20006	. . . . . 6 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
30	29	simplbi 502	. . . . 5 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵)
31	18, 30	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
32	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑋 ∈ 𝐵)
33		eqid 2762	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
34	19, 22, 20, 33	odcong 19675	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋)))
35	28, 32, 9, 10, 34	syl112anc 1401	. 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋)))
36	14, 27, 35	3bitr2d 310	1 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3414 ifcif 4485 class class class wbr 5107 ↦ cmpt 5190 ran crn 5660 ‘cfv 6537 (class class class)co 7416 Fincfn 8955 0cc0 11125 − cmin 11466 ℕ₀cn0 12529 ℤcz 12616 ♯chash 14394 ∥ cdvds 16344 Basecbs 17303 0_gc0g 17526 Grpcgrp 19056 ._gcmg 19189 odcod 19650 CycGrpccyg 20003 ℤRHomczrh 21711 ℤ/nℤczn 21714

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204 ax-mulf 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-oadd 8462 df-omul 8463 df-er 8699 df-ec 8701 df-qs 8705 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-acn 9950 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-rp 13043 df-fz 13562 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-dvds 16345 df-struct 17241 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-ress 17325 df-plusg 17357 df-mulr 17358 df-starv 17359 df-sca 17360 df-vsca 17361 df-ip 17362 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-0g 17528 df-imas 17596 df-qus 17597 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-mhm 18890 df-grp 19059 df-minusg 19060 df-sbg 19061 df-mulg 19190 df-subg 19245 df-nsg 19246 df-eqg 19247 df-ghm 19340 df-od 19654 df-cmn 19908 df-abl 19909 df-cyg 20004 df-mgp 20273 df-rng 20287 df-ur 20320 df-ring 20373 df-cring 20374 df-oppr 20477 df-dvdsr 20497 df-rhm 20612 df-subrng 20707 df-subrg 20731 df-lmod 21045 df-lss 21115 df-lsp 21155 df-sra 21356 df-rgmod 21357 df-lidl 21394 df-rsp 21395 df-2idl 21451 df-cnfld 21585 df-zring 21659 df-zrh 21715 df-zn 21718

This theorem is used by: cygznlem2a 21779 cygznlem3 21781

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