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Theorem cygznlem1 20973

Description: Lemma for cygzn 20977. (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 14256	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
3	2	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ₀)
4		0nn0 12428	. . . . . . 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 2842	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℕ₀)
9		simprl 769	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐾 ∈ ℤ)
10		simprr 771	. . 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 20956	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀)))
14	8, 9, 10, 13	syl3anc 1371	. 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀)))
15		cygzn.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp)
16		cyggrp 19667	. . . . . . 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 2736	. . . . . . 7 ⊢ (od‘𝐺) = (od‘𝐺)
23	19, 20, 21, 22	cyggenod2 19662	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0))
24	17, 18, 23	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0))
25	24, 1	eqtr4di 2794	. . . 4 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = 𝑁)
26	25	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((od‘𝐺)‘𝑋) = 𝑁)
27	26	breq1d 5115	. 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀)))
28	17	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐺 ∈ Grp)
29	19, 20, 21	iscyggen 19657	. . . . . 6 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
30	29	simplbi 498	. . . . 5 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵)
31	18, 30	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
32	31	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑋 ∈ 𝐵)
33		eqid 2736	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
34	19, 22, 20, 33	odcong 19331	. . 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 1374	. 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋)))
36	14, 27, 35	3bitr2d 306	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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3407 ifcif 4486 class class class wbr 5105 ↦ cmpt 5188 ran crn 5634 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 0cc0 11051 − cmin 11385 ℕ₀cn0 12413 ℤcz 12499 ♯chash 14230 ∥ cdvds 16136 Basecbs 17083 0_gc0g 17321 Grpcgrp 18748 ._gcmg 18872 odcod 19306 CycGrpccyg 19654 ℤRHomczrh 20900 ℤ/nℤczn 20903

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-omul 8417 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-fz 13425 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-imas 17390 df-qus 17391 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-od 19310 df-cmn 19564 df-abl 19565 df-cyg 19655 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907

This theorem is referenced by: cygznlem2a 20974 cygznlem3 20976

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