Theorem cycldlenngric 47889

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 47873	. . . . . . . 8 ⊢ (𝐺 ≃𝑔𝑟 𝐻 ↔ (𝐺 GraphIso 𝐻) ≠ ∅)
2		n0rex 4332	. . . . . . . . 9 ⊢ ((𝐺 GraphIso 𝐻) ≠ ∅ → ∃𝑖 ∈ (𝐺 GraphIso 𝐻)𝑖 ∈ (𝐺 GraphIso 𝐻))
3		eqid 2735	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2735	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5		simprll 778	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝐺 ∈ USPGraph)
6		simprlr 779	. . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑗 → ((iEdg‘𝐺)‘(𝑓‘𝑥)) = ((iEdg‘𝐺)‘(𝑓‘𝑗)))
9	8	imaeq2d 6047	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑗 → (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))) = (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗))))
10	9	fveq2d 6879	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑗 → (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))) = (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
11	10	cbvmptv 5225	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) = (𝑗 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
12		cycliswlk 29726	. . . . . . . . . . . . . . 15 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
13	12	ad2antrl 728	. . . . . . . . . . . . . 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 47864	. . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 47872	. . . . . . . . . . . 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 1138	. . . . . . . . . . . . 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 1201	. . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → (♯‘𝑓) = 𝑁)
21	20	3ad2ant1 1133	. . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . 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 3463	. . . . . . . . . . . . . . 15 ⊢ 𝑖 ∈ V
24		vex 3463	. . . . . . . . . . . . . . 15 ⊢ 𝑝 ∈ V
25	23, 24	coex 7924	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∘ 𝑝) ∈ V
26		vex 3463	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
27	26	dmex 7903	. . . . . . . . . . . . . . 15 ⊢ dom 𝑓 ∈ V
28	27	mptex 7214	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) ∈ V
29		breq12 5124	. . . . . . . . . . . . . . . 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 6884	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 3586	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 1465	. . . . . . . . . . 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 3137	. . . . . . . . 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 1934	. . . . 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 5124	. . . . . . 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 6884	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
47	46	adantl 481	. . . . . 6 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
48	45, 47	anbi12d 632	. . . . 5 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ (𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
49	48	cbvex2vw 2040	. . . 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 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2932  ∃wrex 3060  ∅c0 4308   class class class wbr 5119   ↦ cmpt 5201  ^◡ccnv 5653  dom cdm 5654   “ cima 5657   ∘ ccom 5658  ‘cfv 6530  (class class class)co 7403  ♯chash 14346  iEdgciedg 28922  USPGraphcuspgr 29073  Walkscwlks 29522  Cyclesccycls 29713   GraphIso cgrim 47836   ≃_𝑔𝑟 cgric 47837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-oadd 8482  df-er 8717  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9913  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-nn 12239  df-2 12301  df-n0 12500  df-xnn0 12573  df-z 12587  df-uz 12851  df-fz 13523  df-fzo 13670  df-hash 14347  df-word 14530  df-edg 28973  df-uhgr 28983  df-upgr 29007  df-uspgr 29075  df-wlks 29525  df-trls 29618  df-pths 29642  df-cycls 29715  df-grim 47839  df-gric 47842

This theorem is referenced by: (None)