Theorem xmettxlem 13149

Description: Lemma for xmettx 13150. (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 13147	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5		blrn 13052	. . . . . . . 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 294	. . . . . 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 13084	. . . . . . . . . . . . . 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 13084	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
13	3, 12	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Top)
14		mpoexga 6180	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
15	10, 13, 14	syl2anc 409	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
16		rnexg 4869	. . . . . . . . . . . 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 484	. . . . . . . . . 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 12701	. . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑀 ∈ (∞Met‘𝑋))
22		simplrl 525	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑧 ∈ (𝑋 × 𝑌))
23		xp1st 6133	. . . . . . . . . . . . 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 526	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑝 ∈ ℝ*)
26	8	blopn 13130	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ 𝑝 ∈ ℝ*) → ((1st ‘𝑧)(ball‘𝑀)𝑝) ∈ 𝐽)
27	21, 24, 25, 26	syl3anc 1228	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑁 ∈ (∞Met‘𝑌))
29		xp2nd 6134	. . . . . . . . . . . . 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 13130	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd ‘𝑧) ∈ 𝑌 ∧ 𝑝 ∈ ℝ*) → ((2nd ‘𝑧)(ball‘𝑁)𝑝) ∈ 𝐾)
32	28, 30, 25, 31	syl3anc 1228	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ((2nd ‘𝑧)(ball‘𝑁)𝑝) ∈ 𝐾)
33		simpr 109	. . . . . . . . . . . 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 13148	. . . . . . . . . . . 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 2198	. . . . . . . . . . 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 4618	. . . . . . . . . . . . 13 ⊢ (𝑟 = ((1st ‘𝑧)(ball‘𝑀)𝑝) → (𝑟 × 𝑠) = (((1st ‘𝑧)(ball‘𝑀)𝑝) × 𝑠))
37	36	eqeq2d 2177	. . . . . . . . . . . 12 ⊢ (𝑟 = ((1st ‘𝑧)(ball‘𝑀)𝑝) → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (((1st ‘𝑧)(ball‘𝑀)𝑝) × 𝑠)))
38		xpeq2 4619	. . . . . . . . . . . . 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 2177	. . . . . . . . . . . 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 2845	. . . . . . . . . . 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 1228	. . . . . . . . . 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 2165	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
43	42	elrnmpog 5954	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠)))
44	43	elv 2730	. . . . . . . . . 10 ⊢ (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠))
45	41, 44	sylibr 133	. . . . . . . . 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 3143	. . . . . . . 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 114	. . . . . . 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 2591	. . . . . 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 114	. . . 4 ⊢ (𝜑 → (𝑤 ∈ ran (ball‘𝑃) → 𝑤 ∈ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)))))
51	50	ssrdv 3148	. . 3 ⊢ (𝜑 → ran (ball‘𝑃) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
52		blex 13027	. . . . 5 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃) ∈ V)
53		rnexg 4869	. . . . 5 ⊢ ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
54	4, 52, 53	3syl 17	. . . 4 ⊢ (𝜑 → ran (ball‘𝑃) ∈ V)
55		tgss3 12718	. . . 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 409	. . 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 166	. 2 ⊢ (𝜑 → (topGen‘ran (ball‘𝑃)) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
58		xmettx.l	. . . 4 ⊢ 𝐿 = (MetOpen‘𝑃)
59	58	mopnval 13082	. . 3 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → 𝐿 = (topGen‘ran (ball‘𝑃)))
60	4, 59	syl 14	. 2 ⊢ (𝜑 → 𝐿 = (topGen‘ran (ball‘𝑃)))
61		eqid 2165	. . . 4 ⊢ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
62	61	txval 12895	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
63	10, 13, 62	syl2anc 409	. 2 ⊢ (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
64	57, 60, 63	3sstr4d 3187	1 ⊢ (𝜑 → 𝐿 ⊆ (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ∃wrex 2445  Vcvv 2726   ⊆ wss 3116  {cpr 3577   × cxp 4602  ran crn 4605  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844  1^st c1st 6106  2^nd c2nd 6107  supcsup 6947  ℝ^*cxr 7932   < clt 7933  topGenctg 12571  ∞Metcxmet 12620  ballcbl 12622  MetOpencmopn 12625  Topctop 12635   ×_t ctx 12892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-map 6616  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-tx 12893

This theorem is referenced by:  xmettx  13150