upgrimpthslem2 - Mathbox for Alexander van der Vekens

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Theorem upgrimpthslem2 48384

Description: Lemma 2 for upgrimpths 48385. (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 2736	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2736	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
4	2, 3	grimf1o 48360	. . . . . 6 ⊢ (𝑁 ∈ (𝐺 GraphIso 𝐻) → 𝑁:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
5		f1of1 6779	. . . . . 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 29793	. . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
10	2	wlkp 29685	. . . . . . . . 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 13644	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
13		fzossfz 13633	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
14	12, 13	sstri 3931	. . . . . . . . . 10 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
15	14	sseli 3917	. . . . . . . . 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 7037	. . . . . . 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 29684	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
22		0elfz 13578	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
23	21, 22	syl 17	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 0 ∈ (0...(♯‘𝐹)))
24	10, 23	ffvelcdmd 7037	. . . . . 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 13622	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 1 ≤ 𝑋)
33		elfzoelz 13613	. . . . . . . . 9 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ∈ ℤ)
34		zgt0ge1 12583	. . . . . . . . 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 11714	. . . . . . . . 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 29795	. . . . 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 7214	. . . 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 13579	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
47	21, 46	sylib 218	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
48	10, 47	ffvelcdmd 7037	. . . . . 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 12633	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ∈ ℝ)
54		elfzolt2 13623	. . . . . . 7 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 < (♯‘𝐹))
55	53, 54	ltned 11282	. . . . . 6 ⊢ (𝑋 ∈ (1..^(♯‘𝐹)) → 𝑋 ≠ (♯‘𝐹))
56	55	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → 𝑋 ≠ (♯‘𝐹))
57		pthdivtx 29795	. . . . 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 7214	. . . 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 6940	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((𝑁 ∘ 𝑃)‘𝑋) = (𝑁‘(𝑃‘𝑋)))
65	62, 31	fvco3d 6940	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((𝑁 ∘ 𝑃)‘0) = (𝑁‘(𝑃‘0)))
66	64, 65	neeq12d 2993	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ↔ (𝑁‘(𝑃‘𝑋)) ≠ (𝑁‘(𝑃‘0))))
67	62, 52	fvco3d 6940	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (1..^(♯‘𝐹))) → ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) = (𝑁‘(𝑃‘(♯‘𝐹))))
68	64, 67	neeq12d 2993	. . . 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 2933	. . 3 ⊢ (((𝑁 ∘ 𝑃)‘𝑋) ≠ ((𝑁 ∘ 𝑃)‘0) ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑋) = ((𝑁 ∘ 𝑃)‘0))
72		df-ne 2933	. . 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 2932 class class class wbr 5085 ↦ cmpt 5166 ^◡ccnv 5630 dom cdm 5631 “ cima 5634 ∘ ccom 5635 ⟶wf 6494 –1-1→wf1 6495 –1-1-onto→wf1o 6497 ‘cfv 6498 (class class class)co 7367 0cc0 11038 1c1 11039 < clt 11179 ≤ cle 11180 ℕ₀cn0 12437 ℤcz 12524 ...cfz 13461 ..^cfzo 13608 ♯chash 14292 Vtxcvtx 29065 iEdgciedg 29066 USPGraphcuspgr 29217 Walkscwlks 29665 Pathscpths 29778 GraphIso cgrim 48351

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-hash 14293 df-word 14476 df-wlks 29668 df-trls 29759 df-pths 29782 df-grim 48354

This theorem is referenced by: upgrimpths 48385

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