upgrimpthslem2 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > upgrimpthslem2		Structured version Visualization version GIF version

Theorem upgrimpthslem2 48701

Description: Lemma 2 for upgrimpths 48702. (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 2763	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2763	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
4	2, 3	grimf1o 48677	. . . . . 6 ⊢ (𝑁 ∈ (𝐺 GraphIso 𝐻) → 𝑁:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
5		f1of1 6819	. . . . . 6 ⊢ (𝑁:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → 𝑁:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
6	1, 4, 5	3syl 19	. . . . 5 ⊢ (𝜑 → 𝑁:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
7	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑁:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
8		upgrimpths.p	. . . . . 6 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)
9		pthiswlk 30083	. . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
10	2	wlkp 29975	. . . . . . . . 9 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
11	10	adantr 485	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
12		fzo0ss1 13723	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
13		fzossfz 13712	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
14	12, 13	sstri 3946	. . . . . . . . . 10 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
15	14	sseli 3933	. . . . . . . . 9 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ∈ (0...(♯‘𝐹)))
16	15	adantl 486	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ∈ (0...(♯‘𝐹)))
17	11, 16	ffvelcdmd 7080	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝑋) ∈ (Vtx‘𝐺))
18	17	ex 417	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑋 ∈ (1..^(♯‘𝐹)) → (𝑃‘𝑋) ∈ (Vtx‘𝐺)))
19	8, 9, 18	3syl 19	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (1..^(♯‘𝐹)) → (𝑃‘𝑋) ∈ (Vtx‘𝐺)))
20	19	imp 411	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝑋) ∈ (Vtx‘𝐺))
21		wlkcl 29974	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
22		0elfz 13657	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
23	21, 22	syl 18	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 0 ∈ (0...(♯‘𝐹)))
24	10, 23	ffvelcdmd 7080	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘0) ∈ (Vtx‘𝐺))
25	8, 9, 24	3syl 19	. . . . 5 ⊢ (𝜑 → (𝑃‘0) ∈ (Vtx‘𝐺))
26	25	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘0) ∈ (Vtx‘𝐺))
27	8	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝐹(Paths‘𝐺)𝑃)
28		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ∈ (1..^(♯‘𝐹)))
29	8, 9, 21	3syl 19	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
30	29, 22	syl 18	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...(♯‘𝐹)))
31	30	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 0 ∈ (0...(♯‘𝐹)))
32		elfzole1 13701	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 1 ≤ 𝑋)
33		elfzoelz 13692	. . . . . . . . 9 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ∈ ℤ)
34		zgt0ge1 12654	. . . . . . . . 9 ⊢ (𝑋 ∈ ℤ → (0 < 𝑋 ↔ 1 ≤ 𝑋))
35	33, 34	syl 18	. . . . . . . 8 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → (0 < 𝑋 ↔ 1 ≤ 𝑋))
36		simpr 489	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (1..^(♯‘𝐹)) ∧ 0 < 𝑋) → 0 < 𝑋)
37	36	gt0ne0d 11782	. . . . . . . . 9 ⊢ ((𝑋 ∈ (1..^(♯‘𝐹)) ∧ 0 < 𝑋) → 𝑋 ≠ 0)
38	37	ex 417	. . . . . . . 8 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → (0 < 𝑋 → 𝑋 ≠ 0))
39	35, 38	sylbird 263	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → (1 ≤ 𝑋 → 𝑋 ≠ 0))
40	32, 39	mpd 16	. . . . . 6 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ≠ 0)
41	40	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ≠ 0)
42		pthdivtx 30085	. . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑋 ∈ (1..^(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹)) ∧ 𝑋 ≠ 0)) → (𝑃‘𝑋) ≠ (𝑃‘0))
43	27, 28, 31, 41, 42	syl13anc 1399	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝑋) ≠ (𝑃‘0))
44		dff14i 7257	. . . 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 1399	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘0)))
46		nn0fz0 13658	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
47	21, 46	sylib 221	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
48	10, 47	ffvelcdmd 7080	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃‘(♯‘𝐹)) ∈ (Vtx‘𝐺))
49	8, 9, 48	3syl 19	. . . . 5 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) ∈ (Vtx‘𝐺))
50	49	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘(♯‘𝐹)) ∈ (Vtx‘𝐺))
51	29, 46	sylib 221	. . . . . 6 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
52	51	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
53	33	zred 12704	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ∈ ℝ)
54		elfzolt2 13702	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 < (♯‘𝐹))
55	53, 54	ltned 11350	. . . . . 6 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ≠ (♯‘𝐹))
56	55	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ≠ (♯‘𝐹))
57		pthdivtx 30085	. . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑋 ∈ (1..^(♯‘𝐹)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐹)) ∧ 𝑋 ≠ (♯‘𝐹))) → (𝑃‘𝑋) ≠ (𝑃‘(♯‘𝐹)))
58	27, 28, 52, 56, 57	syl13anc 1399	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑃‘𝑋) ≠ (𝑃‘(♯‘𝐹)))
59		dff14i 7257	. . . 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 1399	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘(♯‘𝐹))))
61	8, 9, 10	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
62	61	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
63	15	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ∈ (0...(♯‘𝐹)))
64	62, 63	fvco3d 6982	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((𝑁 ∘ 𝑃)‘𝑋) = (𝑁‘(𝑃‘𝑋)))
65	62, 31	fvco3d 6982	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((𝑁 ∘ 𝑃)‘0) = (𝑁‘(𝑃‘0)))
66	64, 65	neeq12d 3019	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ↔ (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘0))))
67	62, 52	fvco3d 6982	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) = (𝑁‘(𝑃‘(♯‘𝐹))))
68	64, 67	neeq12d 3019	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) ↔ (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘(♯‘𝐹)))))
69	66, 68	anbi12d 643	. . 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 725	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ∧ ((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
71		df-ne 2959	. . 3 ⊢ (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘0))
72		df-ne 2959	. . 3 ⊢ (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))
73	71, 72	anbi12i 639	. 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 221	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 class class class wbr 5109 ↦ cmpt 5192 ^◡ccnv 5660 dom cdm 5661 “ cima 5664 ∘ ccom 5665 ⟶wf 6532 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 0cc0 11104 1c1 11105 < clt 11247 ≤ cle 11248 ℕ₀cn0 12508 ℤcz 12595 ...cfz 13539 ..^cfzo 13687 ♯chash 14371 Vtxcvtx 29355 iEdgciedg 29356 USPGraphcuspgr 29507 Walkscwlks 29955 Pathscpths 30068 GraphIso cgrim 48668

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-n0 12509 df-z 12596 df-uz 12867 df-fz 13540 df-fzo 13688 df-hash 14372 df-word 14556 df-wlks 29958 df-trls 30049 df-pths 30072 df-grim 48671

This theorem is used by: upgrimpths 48702

W3C validator