xrge0pluscn - Mathbox for Thierry Arnoux

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Theorem xrge0pluscn 33939

Description: The addition operation of the extended nonnegative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)

**Hypotheses**
Ref	Expression
xrge0iifhmeo.1	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
xrge0iifhmeo.k	⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
xrge0pluscn.1	⊢ + = ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))

**Assertion**
Ref	Expression
xrge0pluscn	⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)

Distinct variable group: 𝑥,𝐹 Allowed substitution hints: + (𝑥) 𝐽(𝑥)

Proof of Theorem xrge0pluscn Dummy variables 𝑦 𝑢 𝑣 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrge0iifhmeo.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
2		xrge0iifhmeo.k	. . 3 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
3	1, 2	xrge0iifhmeo 33935	. 2 ⊢ 𝐹 ∈ (IIHomeo𝐽)
4		unitsscn 13540	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
5		xpss12 5700	. . . . 5 ⊢ (((0[,]1) ⊆ ℂ ∧ (0[,]1) ⊆ ℂ) → ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ))
6	4, 4, 5	mp2an 692	. . . 4 ⊢ ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ)
7		ax-mulf 11235	. . . . 5 ⊢ · :(ℂ × ℂ)⟶ℂ
8		ffn 6736	. . . . 5 ⊢ ( · :(ℂ × ℂ)⟶ℂ → · Fn (ℂ × ℂ))
9		fnssresb 6690	. . . . 5 ⊢ ( · Fn (ℂ × ℂ) → (( · ↾ ((0[,]1) × (0[,]1))) Fn ((0[,]1) × (0[,]1)) ↔ ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ)))
10	7, 8, 9	mp2b 10	. . . 4 ⊢ (( · ↾ ((0[,]1) × (0[,]1))) Fn ((0[,]1) × (0[,]1)) ↔ ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ))
11	6, 10	mpbir 231	. . 3 ⊢ ( · ↾ ((0[,]1) × (0[,]1))) Fn ((0[,]1) × (0[,]1))
12		ovres 7599	. . . . 5 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) = (𝑢 · 𝑣))
13		iimulcl 24966	. . . . 5 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢 · 𝑣) ∈ (0[,]1))
14	12, 13	eqeltrd 2841	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1))
15	14	rgen2 3199	. . 3 ⊢ ∀𝑢 ∈ (0[,]1)∀𝑣 ∈ (0[,]1)(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1)
16		ffnov 7559	. . 3 ⊢ (( · ↾ ((0[,]1) × (0[,]1))):((0[,]1) × (0[,]1))⟶(0[,]1) ↔ (( · ↾ ((0[,]1) × (0[,]1))) Fn ((0[,]1) × (0[,]1)) ∧ ∀𝑢 ∈ (0[,]1)∀𝑣 ∈ (0[,]1)(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1)))
17	11, 15, 16	mpbir2an 711	. 2 ⊢ ( · ↾ ((0[,]1) × (0[,]1))):((0[,]1) × (0[,]1))⟶(0[,]1)
18		iccssxr 13470	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
19		xpss12 5700	. . . . . 6 ⊢ (((0[,]+∞) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^) → ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^ × ℝ^*))
20	18, 18, 19	mp2an 692	. . . . 5 ⊢ ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^* × ℝ^*)
21		xaddf 13266	. . . . . 6 ⊢ +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^
22		ffn 6736	. . . . . 6 ⊢ ( +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^ → +_𝑒 Fn (ℝ^* × ℝ^*))
23		fnssresb 6690	. . . . . 6 ⊢ ( +_𝑒 Fn (ℝ^* × ℝ^) → (( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞)) ↔ ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^ × ℝ^*)))
24	21, 22, 23	mp2b 10	. . . . 5 ⊢ (( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞)) ↔ ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^* × ℝ^*))
25	20, 24	mpbir 231	. . . 4 ⊢ ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞))
26		xrge0pluscn.1	. . . . 5 ⊢ + = ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))
27	26	fneq1i 6665	. . . 4 ⊢ ( + Fn ((0[,]+∞) × (0[,]+∞)) ↔ ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞)))
28	25, 27	mpbir 231	. . 3 ⊢ + Fn ((0[,]+∞) × (0[,]+∞))
29	26	oveqi 7444	. . . . 5 ⊢ (𝑎 + 𝑏) = (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏)
30		ovres 7599	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏) = (𝑎 +_𝑒 𝑏))
31		ge0xaddcl 13502	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
32	30, 31	eqeltrd 2841	. . . . 5 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏) ∈ (0[,]+∞))
33	29, 32	eqeltrid 2845	. . . 4 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 + 𝑏) ∈ (0[,]+∞))
34	33	rgen2 3199	. . 3 ⊢ ∀𝑎 ∈ (0[,]+∞)∀𝑏 ∈ (0[,]+∞)(𝑎 + 𝑏) ∈ (0[,]+∞)
35		ffnov 7559	. . 3 ⊢ ( + :((0[,]+∞) × (0[,]+∞))⟶(0[,]+∞) ↔ ( + Fn ((0[,]+∞) × (0[,]+∞)) ∧ ∀𝑎 ∈ (0[,]+∞)∀𝑏 ∈ (0[,]+∞)(𝑎 + 𝑏) ∈ (0[,]+∞)))
36	28, 34, 35	mpbir2an 711	. 2 ⊢ + :((0[,]+∞) × (0[,]+∞))⟶(0[,]+∞)
37		iitopon 24905	. 2 ⊢ II ∈ (TopOn‘(0[,]1))
38		letopon 23213	. . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
39		resttopon 23169	. . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ (0[,]+∞) ⊆ ℝ^) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)))
40	38, 18, 39	mp2an 692	. . 3 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))
41	2, 40	eqeltri 2837	. 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
42	26	oveqi 7444	. . . 4 ⊢ ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣))
43	1	xrge0iifcnv 33932	. . . . . . . 8 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))
44	43	simpli 483	. . . . . . 7 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
45		f1of 6848	. . . . . . 7 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) → 𝐹:(0[,]1)⟶(0[,]+∞))
46	44, 45	ax-mp 5	. . . . . 6 ⊢ 𝐹:(0[,]1)⟶(0[,]+∞)
47	46	ffvelcdmi 7103	. . . . 5 ⊢ (𝑢 ∈ (0[,]1) → (𝐹‘𝑢) ∈ (0[,]+∞))
48	46	ffvelcdmi 7103	. . . . 5 ⊢ (𝑣 ∈ (0[,]1) → (𝐹‘𝑣) ∈ (0[,]+∞))
49		ovres 7599	. . . . 5 ⊢ (((𝐹‘𝑢) ∈ (0[,]+∞) ∧ (𝐹‘𝑣) ∈ (0[,]+∞)) → ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
50	47, 48, 49	syl2an 596	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
51	42, 50	eqtrid 2789	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
52	1, 2	xrge0iifhom 33936	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘(𝑢 · 𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
53	12	eqcomd 2743	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢 · 𝑣) = (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣))
54	53	fveq2d 6910	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘(𝑢 · 𝑣)) = (𝐹‘(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣)))
55	51, 52, 54	3eqtr2rd 2784	. 2 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣)) = ((𝐹‘𝑢) + (𝐹‘𝑣)))
56		eqid 2737	. . . 4 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = ((mulGrp‘ℂ_fld) ↾_s (0[,]1))
57	56	iistmd 33901	. . 3 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ TopMnd
58		cnfldex 21367	. . . . . 6 ⊢ ℂ_fld ∈ V
59		ovex 7464	. . . . . 6 ⊢ (0[,]1) ∈ V
60		eqid 2737	. . . . . . 7 ⊢ (ℂ_fld ↾_s (0[,]1)) = (ℂ_fld ↾_s (0[,]1))
61		eqid 2737	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
62	60, 61	mgpress 20147	. . . . . 6 ⊢ ((ℂ_fld ∈ V ∧ (0[,]1) ∈ V) → ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1))))
63	58, 59, 62	mp2an 692	. . . . 5 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1)))
64	60	dfii4 24910	. . . . 5 ⊢ II = (TopOpen‘(ℂ_fld ↾_s (0[,]1)))
65	63, 64	mgptopn 20145	. . . 4 ⊢ II = (TopOpen‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
66		cnfldbas 21368	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
67	61, 66	mgpbas 20142	. . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
68		cnfldmul 21372	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
69	61, 68	mgpplusg 20141	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
70	7, 8	ax-mp 5	. . . . . 6 ⊢ · Fn (ℂ × ℂ)
71	67, 56, 69, 70, 4	ressplusf 32948	. . . . 5 ⊢ (+_𝑓‘((mulGrp‘ℂ_fld) ↾_s (0[,]1))) = ( · ↾ ((0[,]1) × (0[,]1)))
72	71	eqcomi 2746	. . . 4 ⊢ ( · ↾ ((0[,]1) × (0[,]1))) = (+_𝑓‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
73	65, 72	tmdcn 24091	. . 3 ⊢ (((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ TopMnd → ( · ↾ ((0[,]1) × (0[,]1))) ∈ ((II ×_t II) Cn II))
74	57, 73	ax-mp 5	. 2 ⊢ ( · ↾ ((0[,]1) × (0[,]1))) ∈ ((II ×_t II) Cn II)
75	3, 17, 36, 37, 41, 55, 74	mndpluscn 33925	1 ⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ifcif 4525 ↦ cmpt 5225 × cxp 5683 ^◡ccnv 5684 ↾ cres 5687 Fn wfn 6556 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 +∞cpnf 11292 ℝ^*cxr 11294 ≤ cle 11296 -cneg 11493 +_𝑒 cxad 13152 [,]cicc 13390 expce 16097 ↾_s cress 17274 ↾_t crest 17465 ordTopcordt 17544 +_𝑓cplusf 18650 mulGrpcmgp 20137 ℂ_fldccnfld 21364 TopOnctopon 22916 Cn ccn 23232 ×_t ctx 23568 TopMndctmd 24078 IIcii 24901 logclog 26596

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-ordt 17546 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-ps 18611 df-tsr 18612 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-abv 20810 df-lmod 20860 df-scaf 20861 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tmd 24080 df-tgp 24081 df-trg 24168 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 df-ii 24903 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598

This theorem is referenced by: xrge0tmdALT 33945

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