xrge0pluscn - Mathbox for Thierry Arnoux

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Theorem xrge0pluscn 31792

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 31788	. 2 ⊢ 𝐹 ∈ (IIHomeo𝐽)
4		unitsscn 13161	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
5		xpss12 5595	. . . . 5 ⊢ (((0[,]1) ⊆ ℂ ∧ (0[,]1) ⊆ ℂ) → ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ))
6	4, 4, 5	mp2an 688	. . . 4 ⊢ ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ)
7		ax-mulf 10882	. . . . 5 ⊢ · :(ℂ × ℂ)⟶ℂ
8		ffn 6584	. . . . 5 ⊢ ( · :(ℂ × ℂ)⟶ℂ → · Fn (ℂ × ℂ))
9		fnssresb 6538	. . . . 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 230	. . 3 ⊢ ( · ↾ ((0[,]1) × (0[,]1))) Fn ((0[,]1) × (0[,]1))
12		ovres 7416	. . . . 5 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) = (𝑢 · 𝑣))
13		iimulcl 24006	. . . . 5 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢 · 𝑣) ∈ (0[,]1))
14	12, 13	eqeltrd 2839	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1))
15	14	rgen2 3126	. . 3 ⊢ ∀𝑢 ∈ (0[,]1)∀𝑣 ∈ (0[,]1)(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1)
16		ffnov 7379	. . 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 707	. 2 ⊢ ( · ↾ ((0[,]1) × (0[,]1))):((0[,]1) × (0[,]1))⟶(0[,]1)
18		iccssxr 13091	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
19		xpss12 5595	. . . . . 6 ⊢ (((0[,]+∞) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^) → ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^ × ℝ^*))
20	18, 18, 19	mp2an 688	. . . . 5 ⊢ ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^* × ℝ^*)
21		xaddf 12887	. . . . . 6 ⊢ +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^
22		ffn 6584	. . . . . 6 ⊢ ( +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^ → +_𝑒 Fn (ℝ^* × ℝ^*))
23		fnssresb 6538	. . . . . 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 230	. . . 4 ⊢ ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞))
26		xrge0pluscn.1	. . . . 5 ⊢ + = ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))
27	26	fneq1i 6514	. . . 4 ⊢ ( + Fn ((0[,]+∞) × (0[,]+∞)) ↔ ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞)))
28	25, 27	mpbir 230	. . 3 ⊢ + Fn ((0[,]+∞) × (0[,]+∞))
29	26	oveqi 7268	. . . . 5 ⊢ (𝑎 + 𝑏) = (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏)
30		ovres 7416	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏) = (𝑎 +_𝑒 𝑏))
31		ge0xaddcl 13123	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
32	30, 31	eqeltrd 2839	. . . . 5 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏) ∈ (0[,]+∞))
33	29, 32	eqeltrid 2843	. . . 4 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 + 𝑏) ∈ (0[,]+∞))
34	33	rgen2 3126	. . 3 ⊢ ∀𝑎 ∈ (0[,]+∞)∀𝑏 ∈ (0[,]+∞)(𝑎 + 𝑏) ∈ (0[,]+∞)
35		ffnov 7379	. . 3 ⊢ ( + :((0[,]+∞) × (0[,]+∞))⟶(0[,]+∞) ↔ ( + Fn ((0[,]+∞) × (0[,]+∞)) ∧ ∀𝑎 ∈ (0[,]+∞)∀𝑏 ∈ (0[,]+∞)(𝑎 + 𝑏) ∈ (0[,]+∞)))
36	28, 34, 35	mpbir2an 707	. 2 ⊢ + :((0[,]+∞) × (0[,]+∞))⟶(0[,]+∞)
37		iitopon 23948	. 2 ⊢ II ∈ (TopOn‘(0[,]1))
38		letopon 22264	. . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
39		resttopon 22220	. . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ (0[,]+∞) ⊆ ℝ^) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)))
40	38, 18, 39	mp2an 688	. . 3 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))
41	2, 40	eqeltri 2835	. 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
42	26	oveqi 7268	. . . 4 ⊢ ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣))
43	1	xrge0iifcnv 31785	. . . . . . . 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 6700	. . . . . . 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	ffvelrni 6942	. . . . 5 ⊢ (𝑢 ∈ (0[,]1) → (𝐹‘𝑢) ∈ (0[,]+∞))
48	46	ffvelrni 6942	. . . . 5 ⊢ (𝑣 ∈ (0[,]1) → (𝐹‘𝑣) ∈ (0[,]+∞))
49		ovres 7416	. . . . 5 ⊢ (((𝐹‘𝑢) ∈ (0[,]+∞) ∧ (𝐹‘𝑣) ∈ (0[,]+∞)) → ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
50	47, 48, 49	syl2an 595	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
51	42, 50	syl5eq 2791	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
52	1, 2	xrge0iifhom 31789	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘(𝑢 · 𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
53	12	eqcomd 2744	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢 · 𝑣) = (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣))
54	53	fveq2d 6760	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘(𝑢 · 𝑣)) = (𝐹‘(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣)))
55	51, 52, 54	3eqtr2rd 2785	. 2 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣)) = ((𝐹‘𝑢) + (𝐹‘𝑣)))
56		eqid 2738	. . . 4 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = ((mulGrp‘ℂ_fld) ↾_s (0[,]1))
57	56	iistmd 31754	. . 3 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ TopMnd
58		cnfldex 20513	. . . . . 6 ⊢ ℂ_fld ∈ V
59		ovex 7288	. . . . . 6 ⊢ (0[,]1) ∈ V
60		eqid 2738	. . . . . . 7 ⊢ (ℂ_fld ↾_s (0[,]1)) = (ℂ_fld ↾_s (0[,]1))
61		eqid 2738	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
62	60, 61	mgpress 19650	. . . . . 6 ⊢ ((ℂ_fld ∈ V ∧ (0[,]1) ∈ V) → ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1))))
63	58, 59, 62	mp2an 688	. . . . 5 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1)))
64	60	dfii4 23953	. . . . 5 ⊢ II = (TopOpen‘(ℂ_fld ↾_s (0[,]1)))
65	63, 64	mgptopn 19647	. . . 4 ⊢ II = (TopOpen‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
66		cnfldbas 20514	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
67	61, 66	mgpbas 19641	. . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
68		cnfldmul 20516	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
69	61, 68	mgpplusg 19639	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
70	7, 8	ax-mp 5	. . . . . 6 ⊢ · Fn (ℂ × ℂ)
71	67, 56, 69, 70, 4	ressplusf 31137	. . . . 5 ⊢ (+_𝑓‘((mulGrp‘ℂ_fld) ↾_s (0[,]1))) = ( · ↾ ((0[,]1) × (0[,]1)))
72	71	eqcomi 2747	. . . 4 ⊢ ( · ↾ ((0[,]1) × (0[,]1))) = (+_𝑓‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
73	65, 72	tmdcn 23142	. . 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 31778	1 ⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 Vcvv 3422 ⊆ wss 3883 ifcif 4456 ↦ cmpt 5153 × cxp 5578 ^◡ccnv 5579 ↾ cres 5582 Fn wfn 6413 ⟶wf 6414 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 0cc0 10802 1c1 10803 · cmul 10807 +∞cpnf 10937 ℝ^*cxr 10939 ≤ cle 10941 -cneg 11136 +_𝑒 cxad 12775 [,]cicc 13011 expce 15699 ↾_s cress 16867 ↾_t crest 17048 ordTopcordt 17127 +_𝑓cplusf 18238 mulGrpcmgp 19635 ℂ_fldccnfld 20510 TopOnctopon 21967 Cn ccn 22283 ×_t ctx 22619 TopMndctmd 23129 IIcii 23944 logclog 25615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-ordt 17129 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-ps 18199 df-tsr 18200 df-plusf 18240 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-subrg 19937 df-abv 19992 df-lmod 20040 df-scaf 20041 df-sra 20349 df-rgmod 20350 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-tmd 23131 df-tgp 23132 df-trg 23219 df-xms 23381 df-ms 23382 df-tms 23383 df-nm 23644 df-ngp 23645 df-nrg 23647 df-nlm 23648 df-ii 23946 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617

This theorem is referenced by: xrge0tmdALT 31798

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