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Theorem cygznlem1 21560

Description: Lemma for cygzn 21564. (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 14313	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
3	2	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ₀)
4		0nn0 12447	. . . . . . 7 ⊢ 0 ∈ ℕ₀
5	4	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ₀)
6	3, 5	ifclda 4503	. . . . 5 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ₀)
7	1, 6	eqeltrid 2841	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℕ₀)
9		simprl 771	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐾 ∈ ℤ)
10		simprr 773	. . 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 21543	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀)))
14	8, 9, 10, 13	syl3anc 1374	. 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀)))
15		cygzn.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp)
16		cyggrp 19860	. . . . . . 7 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
17	15, 16	syl 17	. . . . . 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 2737	. . . . . . 7 ⊢ (od‘𝐺) = (od‘𝐺)
23	19, 20, 21, 22	cyggenod2 19855	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0))
24	17, 18, 23	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0))
25	24, 1	eqtr4di 2790	. . . 4 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = 𝑁)
26	25	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((od‘𝐺)‘𝑋) = 𝑁)
27	26	breq1d 5096	. 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀)))
28	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐺 ∈ Grp)
29	19, 20, 21	iscyggen 19850	. . . . . 6 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
30	29	simplbi 496	. . . . 5 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵)
31	18, 30	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑋 ∈ 𝐵)
33		eqid 2737	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
34	19, 22, 20, 33	odcong 19519	. . 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 1377	. 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋)))
36	14, 27, 35	3bitr2d 307	1 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3390 ifcif 4467 class class class wbr 5086 ↦ cmpt 5167 ran crn 5627 ‘cfv 6494 (class class class)co 7362 Fincfn 8888 0cc0 11033 − cmin 11372 ℕ₀cn0 12432 ℤcz 12519 ♯chash 14287 ∥ cdvds 16216 Basecbs 17174 0_gc0g 17397 Grpcgrp 18904 ._gcmg 19038 odcod 19494 CycGrpccyg 19847 ℤRHomczrh 21493 ℤ/nℤczn 21496

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-oadd 8404 df-omul 8405 df-er 8638 df-ec 8640 df-qs 8644 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-sup 9350 df-inf 9351 df-oi 9420 df-card 9858 df-acn 9861 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-dvds 16217 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-0g 17399 df-imas 17467 df-qus 17468 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-nsg 19095 df-eqg 19096 df-ghm 19183 df-od 19498 df-cmn 19752 df-abl 19753 df-cyg 19848 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-rhm 20447 df-subrng 20518 df-subrg 20542 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21164 df-rgmod 21165 df-lidl 21202 df-rsp 21203 df-2idl 21244 df-cnfld 21349 df-zring 21441 df-zrh 21497 df-zn 21500

This theorem is referenced by: cygznlem2a 21561 cygznlem3 21563

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