Theorem xmettxlem 14745

Description: Lemma for xmettx 14746. (Contributed by Jim Kingdon, 15-Oct-2023.)

Hypotheses

Ref	Expression
xmetxp.p	⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))
xmetxp.1	⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
xmetxp.2	⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
xmettx.j	⊢ 𝐽 = (MetOpen‘𝑀)
xmettx.k	⊢ 𝐾 = (MetOpen‘𝑁)
xmettx.l	⊢ 𝐿 = (MetOpen‘𝑃)

Assertion

Ref	Expression
xmettxlem	⊢ (𝜑 → 𝐿 ⊆ (𝐽 ×t 𝐾))

Distinct variable groups:   𝑢,𝑀,𝑣   𝑢,𝑁,𝑣   𝑢,𝑋,𝑣   𝑢,𝑌,𝑣

Allowed substitution hints:   𝜑(𝑣,𝑢)   𝑃(𝑣,𝑢)   𝐽(𝑣,𝑢)   𝐾(𝑣,𝑢)   𝐿(𝑣,𝑢)

Proof of Theorem xmettxlem

Dummy variables 𝑝 𝑟 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetxp.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))
2		xmetxp.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
3		xmetxp.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
4	1, 2, 3	xmetxp 14743	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5		blrn 14648	. . . . . . . 8 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (𝑤 ∈ ran (ball‘𝑃) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)∃𝑝 ∈ ℝ* 𝑤 = (𝑧(ball‘𝑃)𝑝)))
6	4, 5	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ran (ball‘𝑃) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)∃𝑝 ∈ ℝ* 𝑤 = (𝑧(ball‘𝑃)𝑝)))
7	6	biimpa 296	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) → ∃𝑧 ∈ (𝑋 × 𝑌)∃𝑝 ∈ ℝ* 𝑤 = (𝑧(ball‘𝑃)𝑝))
8		xmettx.j	. . . . . . . . . . . . . . 15 ⊢ 𝐽 = (MetOpen‘𝑀)
9	8	mopntop 14680	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
10	2, 9	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
11		xmettx.k	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = (MetOpen‘𝑁)
12	11	mopntop 14680	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
13	3, 12	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Top)
14		mpoexga 6270	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
15	10, 13, 14	syl2anc 411	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
16		rnexg 4931	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V → ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
17	15, 16	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
18	17	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
19		bastg 14297	. . . . . . . . . 10 ⊢ (ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V → ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
20	18, 19	syl 14	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
21	2	ad3antrrr 492	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑀 ∈ (∞Met‘𝑋))
22		simplrl 535	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑧 ∈ (𝑋 × 𝑌))
23		xp1st 6223	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (1st ‘𝑧) ∈ 𝑋)
24	22, 23	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → (1st ‘𝑧) ∈ 𝑋)
25		simplrr 536	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑝 ∈ ℝ*)
26	8	blopn 14726	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ 𝑝 ∈ ℝ*) → ((1st ‘𝑧)(ball‘𝑀)𝑝) ∈ 𝐽)
27	21, 24, 25, 26	syl3anc 1249	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ((1st ‘𝑧)(ball‘𝑀)𝑝) ∈ 𝐽)
28	3	ad3antrrr 492	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑁 ∈ (∞Met‘𝑌))
29		xp2nd 6224	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (2nd ‘𝑧) ∈ 𝑌)
30	22, 29	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → (2nd ‘𝑧) ∈ 𝑌)
31	11	blopn 14726	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd ‘𝑧) ∈ 𝑌 ∧ 𝑝 ∈ ℝ*) → ((2nd ‘𝑧)(ball‘𝑁)𝑝) ∈ 𝐾)
32	28, 30, 25, 31	syl3anc 1249	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ((2nd ‘𝑧)(ball‘𝑁)𝑝) ∈ 𝐾)
33		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑤 = (𝑧(ball‘𝑃)𝑝))
34	1, 21, 28, 25, 22	xmetxpbl 14744	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → (𝑧(ball‘𝑃)𝑝) = (((1st ‘𝑧)(ball‘𝑀)𝑝) × ((2nd ‘𝑧)(ball‘𝑁)𝑝)))
35	33, 34	eqtrd 2229	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑤 = (((1st ‘𝑧)(ball‘𝑀)𝑝) × ((2nd ‘𝑧)(ball‘𝑁)𝑝)))
36		xpeq1 4677	. . . . . . . . . . . . 13 ⊢ (𝑟 = ((1st ‘𝑧)(ball‘𝑀)𝑝) → (𝑟 × 𝑠) = (((1st ‘𝑧)(ball‘𝑀)𝑝) × 𝑠))
37	36	eqeq2d 2208	. . . . . . . . . . . 12 ⊢ (𝑟 = ((1st ‘𝑧)(ball‘𝑀)𝑝) → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (((1st ‘𝑧)(ball‘𝑀)𝑝) × 𝑠)))
38		xpeq2 4678	. . . . . . . . . . . . 13 ⊢ (𝑠 = ((2nd ‘𝑧)(ball‘𝑁)𝑝) → (((1st ‘𝑧)(ball‘𝑀)𝑝) × 𝑠) = (((1st ‘𝑧)(ball‘𝑀)𝑝) × ((2nd ‘𝑧)(ball‘𝑁)𝑝)))
39	38	eqeq2d 2208	. . . . . . . . . . . 12 ⊢ (𝑠 = ((2nd ‘𝑧)(ball‘𝑁)𝑝) → (𝑤 = (((1st ‘𝑧)(ball‘𝑀)𝑝) × 𝑠) ↔ 𝑤 = (((1st ‘𝑧)(ball‘𝑀)𝑝) × ((2nd ‘𝑧)(ball‘𝑁)𝑝))))
40	37, 39	rspc2ev 2883	. . . . . . . . . . 11 ⊢ ((((1st ‘𝑧)(ball‘𝑀)𝑝) ∈ 𝐽 ∧ ((2nd ‘𝑧)(ball‘𝑁)𝑝) ∈ 𝐾 ∧ 𝑤 = (((1st ‘𝑧)(ball‘𝑀)𝑝) × ((2nd ‘𝑧)(ball‘𝑁)𝑝))) → ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠))
41	27, 32, 35, 40	syl3anc 1249	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠))
42		eqid 2196	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
43	42	elrnmpog 6035	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠)))
44	43	elv 2767	. . . . . . . . . 10 ⊢ (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠))
45	41, 44	sylibr 134	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)))
46	20, 45	sseldd 3184	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑤 ∈ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
47	46	ex 115	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) → (𝑤 = (𝑧(ball‘𝑃)𝑝) → 𝑤 ∈ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)))))
48	47	rexlimdvva 2622	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) → (∃𝑧 ∈ (𝑋 × 𝑌)∃𝑝 ∈ ℝ* 𝑤 = (𝑧(ball‘𝑃)𝑝) → 𝑤 ∈ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)))))
49	7, 48	mpd 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) → 𝑤 ∈ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
50	49	ex 115	. . . 4 ⊢ (𝜑 → (𝑤 ∈ ran (ball‘𝑃) → 𝑤 ∈ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)))))
51	50	ssrdv 3189	. . 3 ⊢ (𝜑 → ran (ball‘𝑃) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
52		blex 14623	. . . . 5 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃) ∈ V)
53		rnexg 4931	. . . . 5 ⊢ ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
54	4, 52, 53	3syl 17	. . . 4 ⊢ (𝜑 → ran (ball‘𝑃) ∈ V)
55		tgss3 14314	. . . 4 ⊢ ((ran (ball‘𝑃) ∈ V ∧ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V) → ((topGen‘ran (ball‘𝑃)) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ↔ ran (ball‘𝑃) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)))))
56	54, 17, 55	syl2anc 411	. . 3 ⊢ (𝜑 → ((topGen‘ran (ball‘𝑃)) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ↔ ran (ball‘𝑃) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)))))
57	51, 56	mpbird 167	. 2 ⊢ (𝜑 → (topGen‘ran (ball‘𝑃)) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
58		xmettx.l	. . . 4 ⊢ 𝐿 = (MetOpen‘𝑃)
59	58	mopnval 14678	. . 3 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → 𝐿 = (topGen‘ran (ball‘𝑃)))
60	4, 59	syl 14	. 2 ⊢ (𝜑 → 𝐿 = (topGen‘ran (ball‘𝑃)))
61		eqid 2196	. . . 4 ⊢ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
62	61	txval 14491	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
63	10, 13, 62	syl2anc 411	. 2 ⊢ (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
64	57, 60, 63	3sstr4d 3228	1 ⊢ (𝜑 → 𝐿 ⊆ (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∃wrex 2476  Vcvv 2763   ⊆ wss 3157  {cpr 3623   × cxp 4661  ran crn 4664  ‘cfv 5258  (class class class)co 5922   ∈ cmpo 5924  1^st c1st 6196  2^nd c2nd 6197  supcsup 7048  ℝ^*cxr 8060   < clt 8061  topGenctg 12925  ∞Metcxmet 14092  ballcbl 14094  MetOpencmopn 14097  Topctop 14233   ×_t ctx 14488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-map 6709  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-bl 14102  df-mopn 14103  df-top 14234  df-topon 14247  df-bases 14279  df-tx 14489

This theorem is referenced by:  xmettx  14746