Theorem xmettxlem 15235

Description: Lemma for xmettx 15236. (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 15233	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5		blrn 15138	. . . . . . . 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 15170	. . . . . . . . . . . . . 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 15170	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
13	3, 12	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Top)
14		mpoexga 6377	. . . . . . . . . . . . 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 4997	. . . . . . . . . . . 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 14787	. . . . . . . . . 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 6328	. . . . . . . . . . . . 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 15216	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ 𝑝 ∈ ℝ*) → ((1st ‘𝑧)(ball‘𝑀)𝑝) ∈ 𝐽)
27	21, 24, 25, 26	syl3anc 1273	. . . . . . . . . . 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 6329	. . . . . . . . . . . . 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 15216	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd ‘𝑧) ∈ 𝑌 ∧ 𝑝 ∈ ℝ*) → ((2nd ‘𝑧)(ball‘𝑁)𝑝) ∈ 𝐾)
32	28, 30, 25, 31	syl3anc 1273	. . . . . . . . . . 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 15234	. . . . . . . . . . . 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 4739	. . . . . . . . . . . . 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 4740	. . . . . . . . . . . . 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 2925	. . . . . . . . . . 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 1273	. . . . . . . . . 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 6134	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠)))
44	43	elv 2806	. . . . . . . . . 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 3228	. . . . . . . 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 2658	. . . . . 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 3233	. . 3 ⊢ (𝜑 → ran (ball‘𝑃) ⊆ (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
52		blex 15113	. . . . 5 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃) ∈ V)
53		rnexg 4997	. . . . 5 ⊢ ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
54	4, 52, 53	3syl 17	. . . 4 ⊢ (𝜑 → ran (ball‘𝑃) ∈ V)
55		tgss3 14804	. . . 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 15168	. . 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 14981	. . 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 3272	1 ⊢ (𝜑 → 𝐿 ⊆ (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∃wrex 2511  Vcvv 2802   ⊆ wss 3200  {cpr 3670   × cxp 4723  ran crn 4726  ‘cfv 5326  (class class class)co 6018   ∈ cmpo 6020  1^st c1st 6301  2^nd c2nd 6302  supcsup 7181  ℝ^*cxr 8213   < clt 8214  topGenctg 13338  ∞Metcxmet 14552  ballcbl 14554  MetOpencmopn 14557  Topctop 14723   ×_t ctx 14978

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-seqfrec 10710  df-exp 10801  df-cj 11403  df-re 11404  df-im 11405  df-rsqrt 11559  df-abs 11560  df-topgen 13344  df-psmet 14559  df-xmet 14560  df-bl 14562  df-mopn 14563  df-top 14724  df-topon 14737  df-bases 14769  df-tx 14979

This theorem is referenced by:  xmettx  15236