crctcshlem4 - Metamath Proof Explorer

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Theorem crctcshlem4 29063

Description: Lemma for crctcsh 29067. (Contributed by AV, 10-Mar-2021.)

**Hypotheses**
Ref	Expression
crctcsh.v	⊢ 𝑉 = (Vtx‘𝐺)
crctcsh.i	⊢ 𝐼 = (iEdg‘𝐺)
crctcsh.d	⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)
crctcsh.n	⊢ 𝑁 = (♯‘𝐹)
crctcsh.s	⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))
crctcsh.h	⊢ 𝐻 = (𝐹 cyclShift 𝑆)
crctcsh.q	⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))

**Assertion**
Ref	Expression
crctcshlem4	⊢ ((𝜑 ∧ 𝑆 = 0) → (𝐻 = 𝐹 ∧ 𝑄 = 𝑃))

Distinct variable groups: 𝑥,𝑁 𝑥,𝑃 𝑥,𝑆 𝜑,𝑥 Allowed substitution hints: 𝑄(𝑥) 𝐹(𝑥) 𝐺(𝑥) 𝐻(𝑥) 𝐼(𝑥) 𝑉(𝑥)

**Proof of Theorem crctcshlem4**
Step	Hyp	Ref	Expression
1		crctcsh.h	. . 3 ⊢ 𝐻 = (𝐹 cyclShift 𝑆)
2		oveq2 7413	. . . 4 ⊢ (𝑆 = 0 → (𝐹 cyclShift 𝑆) = (𝐹 cyclShift 0))
3		crctcsh.d	. . . . . 6 ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)
4		crctiswlk 29042	. . . . . 6 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
5		crctcsh.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
6	5	wlkf 28860	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
7	3, 4, 6	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
8		cshw0 14740	. . . . 5 ⊢ (𝐹 ∈ Word dom 𝐼 → (𝐹 cyclShift 0) = 𝐹)
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → (𝐹 cyclShift 0) = 𝐹)
10	2, 9	sylan9eqr 2794	. . 3 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝐹 cyclShift 𝑆) = 𝐹)
11	1, 10	eqtrid 2784	. 2 ⊢ ((𝜑 ∧ 𝑆 = 0) → 𝐻 = 𝐹)
12		crctcsh.q	. . 3 ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
13		oveq2 7413	. . . . . . . . 9 ⊢ (𝑆 = 0 → (𝑁 − 𝑆) = (𝑁 − 0))
14		crctcsh.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺)
15		crctcsh.n	. . . . . . . . . . . 12 ⊢ 𝑁 = (♯‘𝐹)
16	14, 5, 3, 15	crctcshlem1 29060	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
17	16	nn0cnd 12530	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
18	17	subid1d 11556	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 0) = 𝑁)
19	13, 18	sylan9eqr 2794	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝑁 − 𝑆) = 𝑁)
20	19	breq2d 5159	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝑥 ≤ (𝑁 − 𝑆) ↔ 𝑥 ≤ 𝑁))
21	20	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 ≤ (𝑁 − 𝑆) ↔ 𝑥 ≤ 𝑁))
22		oveq2 7413	. . . . . . . . 9 ⊢ (𝑆 = 0 → (𝑥 + 𝑆) = (𝑥 + 0))
23	22	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝑥 + 𝑆) = (𝑥 + 0))
24		elfzelz 13497	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℤ)
25	24	zcnd 12663	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℂ)
26	25	addridd 11410	. . . . . . . 8 ⊢ (𝑥 ∈ (0...𝑁) → (𝑥 + 0) = 𝑥)
27	23, 26	sylan9eq 2792	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 + 𝑆) = 𝑥)
28	27	fveq2d 6892	. . . . . 6 ⊢ (((𝜑 ∧ 𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑃‘(𝑥 + 𝑆)) = (𝑃‘𝑥))
29	27	fvoveq1d 7427	. . . . . 6 ⊢ (((𝜑 ∧ 𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑃‘((𝑥 + 𝑆) − 𝑁)) = (𝑃‘(𝑥 − 𝑁)))
30	21, 28, 29	ifbieq12d 4555	. . . . 5 ⊢ (((𝜑 ∧ 𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) = if(𝑥 ≤ 𝑁, (𝑃‘𝑥), (𝑃‘(𝑥 − 𝑁))))
31	30	mpteq2dva 5247	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ 𝑁, (𝑃‘𝑥), (𝑃‘(𝑥 − 𝑁)))))
32		elfzle2 13501	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑁) → 𝑥 ≤ 𝑁)
33	32	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0...𝑁)) → 𝑥 ≤ 𝑁)
34	33	iftrued 4535	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0...𝑁)) → if(𝑥 ≤ 𝑁, (𝑃‘𝑥), (𝑃‘(𝑥 − 𝑁))) = (𝑃‘𝑥))
35	34	mpteq2dva 5247	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ 𝑁, (𝑃‘𝑥), (𝑃‘(𝑥 − 𝑁)))) = (𝑥 ∈ (0...𝑁) ↦ (𝑃‘𝑥)))
36	14	wlkp 28862	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
37	3, 4, 36	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
38		ffn 6714	. . . . . . . . . . 11 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → 𝑃 Fn (0...(♯‘𝐹)))
39	15	eqcomi 2741	. . . . . . . . . . . . 13 ⊢ (♯‘𝐹) = 𝑁
40	39	oveq2i 7416	. . . . . . . . . . . 12 ⊢ (0...(♯‘𝐹)) = (0...𝑁)
41	40	fneq2i 6644	. . . . . . . . . . 11 ⊢ (𝑃 Fn (0...(♯‘𝐹)) ↔ 𝑃 Fn (0...𝑁))
42	38, 41	sylib 217	. . . . . . . . . 10 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → 𝑃 Fn (0...𝑁))
43	42	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) → 𝑃 Fn (0...𝑁))
44		dffn5 6947	. . . . . . . . 9 ⊢ (𝑃 Fn (0...𝑁) ↔ 𝑃 = (𝑥 ∈ (0...𝑁) ↦ (𝑃‘𝑥)))
45	43, 44	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) → 𝑃 = (𝑥 ∈ (0...𝑁) ↦ (𝑃‘𝑥)))
46	45	eqcomd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) → (𝑥 ∈ (0...𝑁) ↦ (𝑃‘𝑥)) = 𝑃)
47	37, 46	mpdan 685	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0...𝑁) ↦ (𝑃‘𝑥)) = 𝑃)
48	35, 47	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ 𝑁, (𝑃‘𝑥), (𝑃‘(𝑥 − 𝑁)))) = 𝑃)
49	48	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ 𝑁, (𝑃‘𝑥), (𝑃‘(𝑥 − 𝑁)))) = 𝑃)
50	31, 49	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) = 𝑃)
51	12, 50	eqtrid 2784	. 2 ⊢ ((𝜑 ∧ 𝑆 = 0) → 𝑄 = 𝑃)
52	11, 51	jca 512	1 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝐻 = 𝐹 ∧ 𝑄 = 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 0cc0 11106 + caddc 11109 ≤ cle 11245 − cmin 11440 ...cfz 13480 ..^cfzo 13623 ♯chash 14286 Word cword 14460 cyclShift ccsh 14734 Vtxcvtx 28245 iEdgciedg 28246 Walkscwlks 28842 Circuitsccrcts 29030

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-hash 14287 df-word 14461 df-concat 14517 df-substr 14587 df-pfx 14617 df-csh 14735 df-wlks 28845 df-trls 28938 df-crcts 29032

This theorem is referenced by: crctcshwlk 29065 crctcsh 29067

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