Theorem cycldlenngric 49026

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 49010	. . . . . . . 8 ⊢ (𝐺 ≃𝑔𝑟 𝐻 ↔ (𝐺 GraphIso 𝐻) ≠ ∅)
2		n0rex 4304	. . . . . . . . 9 ⊢ ((𝐺 GraphIso 𝐻) ≠ ∅ → ∃𝑖 ∈ (𝐺 GraphIso 𝐻)𝑖 ∈ (𝐺 GraphIso 𝐻))
3		eqid 2760	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2760	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5		simprll 791	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝐺 ∈ USPGraph)
6		simprlr 792	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝐻 ∈ USPGraph)
7		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → 𝑖 ∈ (𝐺 GraphIso 𝐻))
8		2fveq3 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑗 → ((iEdg‘𝐺)‘(𝑓‘𝑥)) = ((iEdg‘𝐺)‘(𝑓‘𝑗)))
9	8	imaeq2d 6051	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑗 → (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))) = (𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗))))
10	9	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑗 → (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥)))) = (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
11	10	cbvmptv 5208	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) = (𝑗 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑗)))))
12		cycliswlk 30386	. . . . . . . . . . . . . . 15 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
13	12	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁)) → 𝑓(Walks‘𝐺)𝑝)
14	13	adantl 487	. . . . . . . . . . . . 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 49001	. . . . . . . . . . . 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 793	. . . . . . . . . . . . 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 49009	. . . . . . . . . . . 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 1156	. . . . . . . . . . . . 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 1219	. . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (𝐺 GraphIso 𝐻) ∧ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁))) → (♯‘𝑓) = 𝑁)
21	20	3ad2ant1 1151	. . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . 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 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑖 ∈ V
24		vex 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑝 ∈ V
25	23, 24	coex 7942	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∘ 𝑝) ∈ V
26		vex 3454	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
27	26	dmex 7921	. . . . . . . . . . . . . . 15 ⊢ dom 𝑓 ∈ V
28	27	mptex 7229	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ dom 𝑓 ↦ (◡(iEdg‘𝐻)‘(𝑖 “ ((iEdg‘𝐺)‘(𝑓‘𝑥))))) ∈ V
29		breq12 5107	. . . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . 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 3561	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 1492	. . . . . . . . . . 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 418	. . . . . . . . . 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 3159	. . . . . . . . 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 18	. . . . . . . 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 220	. . . . . . 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 420	. . . . . 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 1966	. . . . 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 412	. . . 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 5107	. . . . . . 7 ⊢ ((𝑓 = 𝑔 ∧ 𝑝 = 𝑞) → (𝑓(Cycles‘𝐻)𝑝 ↔ 𝑔(Cycles‘𝐻)𝑞))
45	44	ancoms 464	. . . . . 6 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑓(Cycles‘𝐻)𝑝 ↔ 𝑔(Cycles‘𝐻)𝑞))
46		fveqeq2 6894	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
47	46	adantl 487	. . . . . 6 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑔) = 𝑁))
48	45, 47	anbi12d 644	. . . . 5 ⊢ ((𝑝 = 𝑞 ∧ 𝑓 = 𝑔) → ((𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ (𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁)))
49	48	cbvex2vw 2073	. . . 4 ⊢ (∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ ∃𝑞∃𝑔(𝑔(Cycles‘𝐻)𝑞 ∧ (♯‘𝑔) = 𝑁))
50	43, 49	imbitrrdi 255	. . 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 153	. 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 459	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((∃𝑝∃𝑓(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ∧ ¬ ∃𝑝∃𝑓(𝑓(Cycles‘𝐻)𝑝 ∧ (♯‘𝑓) = 𝑁)) → ¬ 𝐺 ≃𝑔𝑟 𝐻))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  ∅c0 4278   class class class wbr 5102   ↦ cmpt 5185  ^◡ccnv 5649  dom cdm 5650   “ cima 5653   ∘ ccom 5654  ‘cfv 6538  (class class class)co 7420  ♯chash 14474  iEdgciedg 29575  USPGraphcuspgr 29729  Walkscwlks 30177  Cyclesccycls 30372   GraphIso cgrim 48973   ≃_𝑔𝑟 cgric 48974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-n0 12607  df-xnn0 12680  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-hash 14475  df-word 14659  df-edg 29626  df-uhgr 29636  df-upgr 29660  df-uspgr 29731  df-wlks 30180  df-trls 30275  df-pths 30299  df-cycls 30374  df-grim 48976  df-gric 48979

This theorem is used by:  gpg5ngric  49226