Theorem xmettxlem 15320

Description: Lemma for xmettx 15321. (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 15318	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5		blrn 15223	. . . . . . . 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 15255	. . . . . . . . . . . . . 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 15255	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
13	3, 12	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Top)
14		mpoexga 6386	. . . . . . . . . . . . 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 5003	. . . . . . . . . . . 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 14872	. . . . . . . . . 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 537	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑧 ∈ (𝑋 × 𝑌))
23		xp1st 6337	. . . . . . . . . . . . 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 538	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑝 ∈ ℝ*)
26	8	blopn 15301	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ 𝑝 ∈ ℝ*) → ((1st ‘𝑧)(ball‘𝑀)𝑝) ∈ 𝐽)
27	21, 24, 25, 26	syl3anc 1274	. . . . . . . . . . 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 6338	. . . . . . . . . . . . 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 15301	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd ‘𝑧) ∈ 𝑌 ∧ 𝑝 ∈ ℝ*) → ((2nd ‘𝑧)(ball‘𝑁)𝑝) ∈ 𝐾)
32	28, 30, 25, 31	syl3anc 1274	. . . . . . . . . . 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 15319	. . . . . . . . . . . 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 2264	. . . . . . . . . . 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 4745	. . . . . . . . . . . . 13 ⊢ (𝑟 = ((1st ‘𝑧)(ball‘𝑀)𝑝) → (𝑟 × 𝑠) = (((1st ‘𝑧)(ball‘𝑀)𝑝) × 𝑠))
37	36	eqeq2d 2243	. . . . . . . . . . . 12 ⊢ (𝑟 = ((1st ‘𝑧)(ball‘𝑀)𝑝) → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (((1st ‘𝑧)(ball‘𝑀)𝑝) × 𝑠)))
38		xpeq2 4746	. . . . . . . . . . . . 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 2243	. . . . . . . . . . . 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 2926	. . . . . . . . . . 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 1274	. . . . . . . . . 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 2231	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
43	42	elrnmpog 6144	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠)))
44	43	elv 2807	. . . . . . . . . 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 3229	. . . . . . . 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 2659	. . . . . 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 3234	. . 3 ⊢ (𝜑 → ran (ball‘𝑃) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
52		blex 15198	. . . . 5 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃) ∈ V)
53		rnexg 5003	. . . . 5 ⊢ ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
54	4, 52, 53	3syl 17	. . . 4 ⊢ (𝜑 → ran (ball‘𝑃) ∈ V)
55		tgss3 14889	. . . 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 15253	. . 3 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → 𝐿 = (topGen‘ran (ball‘𝑃)))
60	4, 59	syl 14	. 2 ⊢ (𝜑 → 𝐿 = (topGen‘ran (ball‘𝑃)))
61		eqid 2231	. . . 4 ⊢ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
62	61	txval 15066	. . 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 3273	1 ⊢ (𝜑 → 𝐿 ⊆ (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∃wrex 2512  Vcvv 2803   ⊆ wss 3201  {cpr 3674   × cxp 4729  ran crn 4732  ‘cfv 5333  (class class class)co 6028   ∈ cmpo 6030  1^st c1st 6310  2^nd c2nd 6311  supcsup 7241  ℝ^*cxr 8272   < clt 8273  topGenctg 13417  ∞Metcxmet 14632  ballcbl 14634  MetOpencmopn 14637  Topctop 14808   ×_t ctx 15063

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-map 6862  df-sup 7243  df-inf 7244  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-xneg 10068  df-xadd 10069  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-topgen 13423  df-psmet 14639  df-xmet 14640  df-bl 14642  df-mopn 14643  df-top 14809  df-topon 14822  df-bases 14854  df-tx 15064

This theorem is referenced by:  xmettx  15321