Theorem cycldlenngric 48433

Description: Two simple pseudographs are not isomorphic if one has a cycle and the other has no cycle of the same length. (Contributed by AV, 6-Nov-2025.)

Assertion

Ref	Expression
cycldlenngric	⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ∧ ¬ ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ¬ 𝐺 ≃𝑔𝑟 𝐻))

Distinct variable groups:   𝑓,𝐺,𝑝   𝑓,𝐻,𝑝   𝑓,𝑁,𝑝

Proof of Theorem cycldlenngric

Dummy variables 𝑔 𝑖 𝑗 𝑞 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgric 48417	. . . . . . . 8 ⊢ (𝐺 ≃𝑔𝑟 𝐻 ↔ (𝐺 GraphIso 𝐻) ≠ ∅)
2		n0rex 4288	. . . . . . . . 9 ⊢ ((𝐺 GraphIso 𝐻) ≠ ∅ → ∃𝑖 ∈ (𝐺 GraphIso 𝐻)𝑖 ∈ (𝐺 GraphIso 𝐻))
3		eqid 2741	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2741	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5		simprll 785	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝐺 ∈ USPGraph)
6		simprlr 786	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝐻 ∈ USPGraph)
7		simpl 484	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝑖 ∈ (𝐺 GraphIso 𝐻))
8		2fveq3 6836	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑗 → ((iEdg‘𝐺)‘(𝑓‘𝑥)) = ((iEdg‘𝐺)‘(𝑓‘𝑗)))
9	8	imaeq2d 6019	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑗 → (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))) = (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗))))
10	9	fveq2d 6835	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑗 → (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))) = (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
11	10	cbvmptv 5179	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) = (𝑗 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
12		cycliswlk 29888	. . . . . . . . . . . . . . 15 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
13	12	ad2antrl 735	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → 𝑓(Walks‘𝐺)𝑝)
14	13	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝑓(Walks‘𝐺)𝑝)
15	3, 4, 5, 6, 7, 11, 14	upgrimwlklen 48408	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)))
16		simprrl 787	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝑓(Cycles‘𝐺)𝑝)
17	3, 4, 5, 6, 7, 11, 16	upgrimcycls 48416	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝))
18		simp3 1145	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝))
19		simp2r 1208	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓))
20		simprrr 788	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → (♯‘𝑓) = 𝑁)
21	20	3ad2ant1 1140	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → (♯‘𝑓) = 𝑁)
22	19, 21	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁)
23		vex 3437	. . . . . . . . . . . . . . 15 ⊢ 𝑖 ∈ V
24		vex 3437	. . . . . . . . . . . . . . 15 ⊢ 𝑝 ∈ V
25	23, 24	coex 7874	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∘ 𝑝) ∈ V
26		vex 3437	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
27	26	dmex 7853	. . . . . . . . . . . . . . 15 ⊢ dom 𝑓 ∈ V
28	27	mptex 7171	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) ∈ V
29		breq12 5080	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) ∧ 𝑞 = (𝑖 ∘ 𝑝)) → (𝑔(Cycles‘𝐻)𝑞 ↔ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)))
30	29	ancoms 460	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑖 ∘ 𝑝) ∧ 𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) → (𝑔(Cycles‘𝐻)𝑞 ↔ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)))
31		fveqeq2 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) → ((♯‘𝑔) = 𝑁 ↔ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁))
32	31	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑖 ∘ 𝑝) ∧ 𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) → ((♯‘𝑔) = 𝑁 ↔ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁))
33	30, 32	anbi12d 639	. . . . . . . . . . . . . 14 ⊢ ((𝑞 = (𝑖 ∘ 𝑝) ∧ 𝑔 = (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) → ((𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁) ↔ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁)))
34	25, 28, 33	spc2ev 3547	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = 𝑁) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
35	18, 22, 34	syl2anc 591	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) ∧ ((𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Walks‘𝐻)(𝑖 ∘ 𝑝) ∧ (♯‘(𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))) = (♯‘𝑓)) ∧ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))))(Cycles‘𝐻)(𝑖 ∘ 𝑝)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
36	15, 17, 35	mpd3an23 1472	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
37	36	ex 414	. . . . . . . . . 10 ⊢ (𝑖 ∈ (𝐺 GraphIso 𝐻) → (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
38	37	rexlimivw 3138	. . . . . . . . 9 ⊢ (∃𝑖 ∈ (𝐺 GraphIso 𝐻)𝑖 ∈ (𝐺 GraphIso 𝐻) → (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
39	2, 38	syl 17	. . . . . . . 8 ⊢ ((𝐺 GraphIso 𝐻) ≠ ∅ → (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
40	1, 39	sylbi 219	. . . . . . 7 ⊢ (𝐺 ≃𝑔𝑟 𝐻 → (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
41	40	expdcom 416	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → (𝐺 ≃𝑔𝑟 𝐻 → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))))
42	41	exlimdvv 1942	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → (𝐺 ≃𝑔𝑟 𝐻 → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))))
43	42	imp 408	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ ∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → (𝐺 ≃𝑔𝑟 𝐻 → ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
44		breq12 5080	. . . . . . 7 ⊢ ((𝑓 = 𝑔 ∧ 𝑝 = 𝑞) → (𝑓(Cycles‘𝐻)𝑝 ↔ 𝑔(Cycles‘𝐻)𝑞))
45	44	ancoms 460	. . . . . 6 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑓(Cycles‘𝐻)𝑝 ↔ 𝑔(Cycles‘𝐻)𝑞))
46		fveqeq2 6840	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
47	46	adantl 483	. . . . . 6 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
48	45, 47	anbi12d 639	. . . . 5 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ (𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
49	48	cbvex2vw 2049	. . . 4 ⊢ (∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
50	43, 49	imbitrrdi 254	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ ∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → (𝐺 ≃𝑔𝑟 𝐻 → ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁)))
51	50	con3d 152	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ ∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → (¬ ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) → ¬ 𝐺 ≃𝑔𝑟 𝐻))
52	51	expimpd 455	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ∧ ¬ ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ¬ 𝐺 ≃𝑔𝑟 𝐻))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548  ∃wex 1787   ∈ wcel 2121   ≠ wne 2936  ∃wrex 3065  ∅c0 4264   class class class wbr 5075   ↦ cmpt 5156  ^◡ccnv 5620  dom cdm 5621   “ cima 5624   ∘ ccom 5625  ‘cfv 6489  (class class class)co 7360  ♯chash 14287  iEdgciedg 29088  USPGraphcuspgr 29239  Walkscwlks 29687  Cyclesccycls 29875   GraphIso cgrim 48380   ≃_𝑔𝑟 cgric 48381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-ifp 1070  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  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-2o 8400  df-oadd 8403  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-n0 12433  df-xnn0 12506  df-z 12520  df-uz 12784  df-fz 13457  df-fzo 13604  df-hash 14288  df-word 14471  df-edg 29139  df-uhgr 29149  df-upgr 29173  df-uspgr 29241  df-wlks 29690  df-trls 29781  df-pths 29804  df-cycls 29877  df-grim 48383  df-gric 48386

This theorem is referenced by:  gpg5ngric  48633