upgrimpthslem2 - Mathbox for Alexander van der Vekens

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Theorem upgrimpthslem2 48221

Description: Lemma 2 for upgrimpths 48222. (Contributed by AV, 31-Oct-2025.)

**Hypotheses**
Ref	Expression
upgrimwlk.i	⊢ 𝐼 = (iEdg‘𝐺)
upgrimwlk.j	⊢ 𝐽 = (iEdg‘𝐻)
upgrimwlk.g	⊢ (𝜑 → 𝐺 ∈ USPGraph)
upgrimwlk.h	⊢ (𝜑 → 𝐻 ∈ USPGraph)
upgrimwlk.n	⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))
upgrimwlk.e	⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (^◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))
upgrimpths.p	⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)

**Assertion**
Ref	Expression
upgrimpthslem2	⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐺 𝑥,𝐼 𝑥,𝐽 𝑥,𝑃 𝜑,𝑥 Allowed substitution hints: 𝐸(𝑥) 𝐻(𝑥) 𝑁(𝑥) 𝑋(𝑥)

**Proof of Theorem upgrimpthslem2**
Step	Hyp	Ref	Expression
1		upgrimwlk.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))
2		eqid 2737	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2737	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
4	2, 3	grimf1o 48197	. . . . . 6 ⊢ (𝑁 ∈ (𝐺 GraphIso 𝐻) → 𝑁:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
5		f1of1 6774	. . . . . 6 ⊢ (𝑁:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → 𝑁:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
6	1, 4, 5	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑁:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑁:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
8		upgrimpths.p	. . . . . 6 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)
9		pthiswlk 29802	. . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
10	2	wlkp 29694	. . . . . . . . 9 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
12		fzo0ss1 13609	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
13		fzossfz 13598	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
14	12, 13	sstri 3944	. . . . . . . . . 10 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
15	14	sseli 3930	. . . . . . . . 9 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ∈ (0...(♯‘𝐹)))
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ∈ (0...(♯‘𝐹)))
17	11, 16	ffvelcdmd 7032	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝑋) ∈ (Vtx‘𝐺))
18	17	ex 412	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑋 ∈ (1..^(♯‘𝐹)) → (𝑃‘𝑋) ∈ (Vtx‘𝐺)))
19	8, 9, 18	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (1..^(♯‘𝐹)) → (𝑃‘𝑋) ∈ (Vtx‘𝐺)))
20	19	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝑋) ∈ (Vtx‘𝐺))
21		wlkcl 29693	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
22		0elfz 13544	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
23	21, 22	syl 17	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 0 ∈ (0...(♯‘𝐹)))
24	10, 23	ffvelcdmd 7032	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘0) ∈ (Vtx‘𝐺))
25	8, 9, 24	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑃‘0) ∈ (Vtx‘𝐺))
26	25	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘0) ∈ (Vtx‘𝐺))
27	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝐹(Paths‘𝐺)𝑃)
28		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ∈ (1..^(♯‘𝐹)))
29	8, 9, 21	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
30	29, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...(♯‘𝐹)))
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 0 ∈ (0...(♯‘𝐹)))
32		elfzole1 13587	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 1 ≤ 𝑋)
33		elfzoelz 13579	. . . . . . . . 9 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ∈ ℤ)
34		zgt0ge1 12550	. . . . . . . . 9 ⊢ (𝑋 ∈ ℤ → (0 < 𝑋 ↔ 1 ≤ 𝑋))
35	33, 34	syl 17	. . . . . . . 8 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → (0 < 𝑋 ↔ 1 ≤ 𝑋))
36		simpr 484	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (1..^(♯‘𝐹)) ∧ 0 < 𝑋) → 0 < 𝑋)
37	36	gt0ne0d 11705	. . . . . . . . 9 ⊢ ((𝑋 ∈ (1..^(♯‘𝐹)) ∧ 0 < 𝑋) → 𝑋 ≠ 0)
38	37	ex 412	. . . . . . . 8 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → (0 < 𝑋 → 𝑋 ≠ 0))
39	35, 38	sylbird 260	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → (1 ≤ 𝑋 → 𝑋 ≠ 0))
40	32, 39	mpd 15	. . . . . 6 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ≠ 0)
41	40	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ≠ 0)
42		pthdivtx 29804	. . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑋 ∈ (1..^(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ 𝑋 ≠ 0)) → (𝑃‘𝑋) ≠ (𝑃‘0))
43	27, 28, 31, 41, 42	syl13anc 1375	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝑋) ≠ (𝑃‘0))
44		dff14i 7207	. . . 4 ⊢ ((𝑁:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) ∧ ((𝑃‘𝑋) ∈ (Vtx‘𝐺) ∧ (𝑃‘0) ∈ (Vtx‘𝐺) ∧ (𝑃‘𝑋) ≠ (𝑃‘0))) → (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘0)))
45	7, 20, 26, 43, 44	syl13anc 1375	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘0)))
46		nn0fz0 13545	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
47	21, 46	sylib 218	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
48	10, 47	ffvelcdmd 7032	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘(♯‘𝐹)) ∈ (Vtx‘𝐺))
49	8, 9, 48	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) ∈ (Vtx‘𝐺))
50	49	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘(♯‘𝐹)) ∈ (Vtx‘𝐺))
51	29, 46	sylib 218	. . . . . 6 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
52	51	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
53	33	zred 12600	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ∈ ℝ)
54		elfzolt2 13588	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 < (♯‘𝐹))
55	53, 54	ltned 11273	. . . . . 6 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ≠ (♯‘𝐹))
56	55	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ≠ (♯‘𝐹))
57		pthdivtx 29804	. . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑋 ∈ (1..^(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹)) ∧ 𝑋 ≠ (♯‘𝐹))) → (𝑃‘𝑋) ≠ (𝑃‘(♯‘𝐹)))
58	27, 28, 52, 56, 57	syl13anc 1375	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝑋) ≠ (𝑃‘(♯‘𝐹)))
59		dff14i 7207	. . . 4 ⊢ ((𝑁:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) ∧ ((𝑃‘𝑋) ∈ (Vtx‘𝐺) ∧ (𝑃‘(♯‘𝐹)) ∈ (Vtx‘𝐺) ∧ (𝑃‘𝑋) ≠ (𝑃‘(♯‘𝐹)))) → (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘(♯‘𝐹))))
60	7, 20, 50, 58, 59	syl13anc 1375	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘(♯‘𝐹))))
61	8, 9, 10	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
62	61	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
63	15	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ∈ (0...(♯‘𝐹)))
64	62, 63	fvco3d 6935	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((𝑁 ∘ 𝑃)‘𝑋) = (𝑁‘(𝑃‘𝑋)))
65	62, 31	fvco3d 6935	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((𝑁 ∘ 𝑃)‘0) = (𝑁‘(𝑃‘0)))
66	64, 65	neeq12d 2994	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ↔ (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘0))))
67	62, 52	fvco3d 6935	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) = (𝑁‘(𝑃‘(♯‘𝐹))))
68	64, 67	neeq12d 2994	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) ↔ (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘(♯‘𝐹)))))
69	66, 68	anbi12d 633	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ∧ ((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) ↔ ((𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘0)) ∧ (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘(♯‘𝐹))))))
70	45, 60, 69	mpbir2and 714	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ∧ ((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
71		df-ne 2934	. . 3 ⊢ (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘0))
72		df-ne 2934	. . 3 ⊢ (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))
73	71, 72	anbi12i 629	. 2 ⊢ ((((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ∧ ((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) ↔ (¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
74	70, 73	sylib 218	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5099 ↦ cmpt 5180 ^◡ccnv 5624 dom cdm 5625 “ cima 5628 ∘ ccom 5629 ⟶wf 6489 –1-1→wf1 6490 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7360 0cc0 11030 1c1 11031 < clt 11170 ≤ cle 11171 ℕ₀cn0 12405 ℤcz 12492 ...cfz 13427 ..^cfzo 13574 ♯chash 14257 Vtxcvtx 29073 iEdgciedg 29074 USPGraphcuspgr 29225 Walkscwlks 29674 Pathscpths 29787 GraphIso cgrim 48188

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 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 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-hash 14258 df-word 14441 df-wlks 29677 df-trls 29768 df-pths 29791 df-grim 48191

This theorem is referenced by: upgrimpths 48222

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