Theorem cycldlenngric 48241

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 48225	. . . . . . . 8 ⊢ (𝐺 ≃𝑔𝑟 𝐻 ↔ (𝐺 GraphIso 𝐻) ≠ ∅)
2		n0rex 4310	. . . . . . . . 9 ⊢ ((𝐺 GraphIso 𝐻) ≠ ∅ → ∃𝑖 ∈ (𝐺 GraphIso 𝐻)𝑖 ∈ (𝐺 GraphIso 𝐻))
3		eqid 2737	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2737	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝐺 ∈ USPGraph)
6		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝐻 ∈ USPGraph)
7		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝑖 ∈ (𝐺 GraphIso 𝐻))
8		2fveq3 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑗 → ((iEdg‘𝐺)‘(𝑓‘𝑥)) = ((iEdg‘𝐺)‘(𝑓‘𝑗)))
9	8	imaeq2d 6020	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑗 → (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))) = (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗))))
10	9	fveq2d 6839	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑗 → (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))) = (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
11	10	cbvmptv 5203	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) = (𝑗 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
12		cycliswlk 29875	. . . . . . . . . . . . . . 15 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
13	12	ad2antrl 729	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → 𝑓(Walks‘𝐺)𝑝)
14	13	adantl 481	. . . . . . . . . . . . 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 48216	. . . . . . . . . . . 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 781	. . . . . . . . . . . . 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 48224	. . . . . . . . . . . 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 1139	. . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → (♯‘𝑓) = 𝑁)
21	20	3ad2ant1 1134	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . 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 3445	. . . . . . . . . . . . . . 15 ⊢ 𝑖 ∈ V
24		vex 3445	. . . . . . . . . . . . . . 15 ⊢ 𝑝 ∈ V
25	23, 24	coex 7874	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∘ 𝑝) ∈ V
26		vex 3445	. . . . . . . . . . . . . . . 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 5104	. . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . 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 3562	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 1466	. . . . . . . . . . 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 412	. . . . . . . . . 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 3134	. . . . . . . . 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 217	. . . . . . 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 414	. . . . . 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 1936	. . . . 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 406	. . . 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 5104	. . . . . . 7 ⊢ ((𝑓 = 𝑔 ∧ 𝑝 = 𝑞) → (𝑓(Cycles‘𝐻)𝑝 ↔ 𝑔(Cycles‘𝐻)𝑞))
45	44	ancoms 458	. . . . . 6 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑓(Cycles‘𝐻)𝑝 ↔ 𝑔(Cycles‘𝐻)𝑞))
46		fveqeq2 6844	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
47	46	adantl 481	. . . . . 6 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
48	45, 47	anbi12d 633	. . . . 5 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ (𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
49	48	cbvex2vw 2043	. . . 4 ⊢ (∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
50	43, 49	imbitrrdi 252	. . 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 453	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3061  ∅c0 4286   class class class wbr 5099   ↦ cmpt 5180  ^◡ccnv 5624  dom cdm 5625   “ cima 5628   ∘ ccom 5629  ‘cfv 6493  (class class class)co 7360  ♯chash 14257  iEdgciedg 29074  USPGraphcuspgr 29225  Walkscwlks 29674  Cyclesccycls 29862   GraphIso cgrim 48188   ≃_𝑔𝑟 cgric 48189

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-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 9817  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-2 12212  df-n0 12406  df-xnn0 12479  df-z 12493  df-uz 12756  df-fz 13428  df-fzo 13575  df-hash 14258  df-word 14441  df-edg 29125  df-uhgr 29135  df-upgr 29159  df-uspgr 29227  df-wlks 29677  df-trls 29768  df-pths 29791  df-cycls 29864  df-grim 48191  df-gric 48194

This theorem is referenced by:  gpg5ngric  48441