xrge0pluscn - Mathbox for Thierry Arnoux

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Theorem xrge0pluscn 31890

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 31886	. 2 ⊢ 𝐹 ∈ (IIHomeo𝐽)
4		unitsscn 13232	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
5		xpss12 5604	. . . . 5 ⊢ (((0[,]1) ⊆ ℂ ∧ (0[,]1) ⊆ ℂ) → ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ))
6	4, 4, 5	mp2an 689	. . . 4 ⊢ ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ)
7		ax-mulf 10951	. . . . 5 ⊢ · :(ℂ × ℂ)⟶ℂ
8		ffn 6600	. . . . 5 ⊢ ( · :(ℂ × ℂ)⟶ℂ → · Fn (ℂ × ℂ))
9		fnssresb 6554	. . . . 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 7438	. . . . 5 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) = (𝑢 · 𝑣))
13		iimulcl 24100	. . . . 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 3120	. . 3 ⊢ ∀𝑢 ∈ (0[,]1)∀𝑣 ∈ (0[,]1)(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1)
16		ffnov 7401	. . 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 708	. 2 ⊢ ( · ↾ ((0[,]1) × (0[,]1))):((0[,]1) × (0[,]1))⟶(0[,]1)
18		iccssxr 13162	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
19		xpss12 5604	. . . . . 6 ⊢ (((0[,]+∞) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^) → ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^ × ℝ^*))
20	18, 18, 19	mp2an 689	. . . . 5 ⊢ ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^* × ℝ^*)
21		xaddf 12958	. . . . . 6 ⊢ +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^
22		ffn 6600	. . . . . 6 ⊢ ( +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^ → +_𝑒 Fn (ℝ^* × ℝ^*))
23		fnssresb 6554	. . . . . 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 6530	. . . 4 ⊢ ( + Fn ((0[,]+∞) × (0[,]+∞)) ↔ ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞)))
28	25, 27	mpbir 230	. . 3 ⊢ + Fn ((0[,]+∞) × (0[,]+∞))
29	26	oveqi 7288	. . . . 5 ⊢ (𝑎 + 𝑏) = (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏)
30		ovres 7438	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏) = (𝑎 +_𝑒 𝑏))
31		ge0xaddcl 13194	. . . . . 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 3120	. . 3 ⊢ ∀𝑎 ∈ (0[,]+∞)∀𝑏 ∈ (0[,]+∞)(𝑎 + 𝑏) ∈ (0[,]+∞)
35		ffnov 7401	. . 3 ⊢ ( + :((0[,]+∞) × (0[,]+∞))⟶(0[,]+∞) ↔ ( + Fn ((0[,]+∞) × (0[,]+∞)) ∧ ∀𝑎 ∈ (0[,]+∞)∀𝑏 ∈ (0[,]+∞)(𝑎 + 𝑏) ∈ (0[,]+∞)))
36	28, 34, 35	mpbir2an 708	. 2 ⊢ + :((0[,]+∞) × (0[,]+∞))⟶(0[,]+∞)
37		iitopon 24042	. 2 ⊢ II ∈ (TopOn‘(0[,]1))
38		letopon 22356	. . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
39		resttopon 22312	. . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ (0[,]+∞) ⊆ ℝ^) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)))
40	38, 18, 39	mp2an 689	. . 3 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))
41	2, 40	eqeltri 2835	. 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
42	26	oveqi 7288	. . . 4 ⊢ ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣))
43	1	xrge0iifcnv 31883	. . . . . . . 8 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))
44	43	simpli 484	. . . . . . 7 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
45		f1of 6716	. . . . . . 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 6960	. . . . 5 ⊢ (𝑢 ∈ (0[,]1) → (𝐹‘𝑢) ∈ (0[,]+∞))
48	46	ffvelrni 6960	. . . . 5 ⊢ (𝑣 ∈ (0[,]1) → (𝐹‘𝑣) ∈ (0[,]+∞))
49		ovres 7438	. . . . 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 2790	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
52	1, 2	xrge0iifhom 31887	. . 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 6778	. . 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 31852	. . 3 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ TopMnd
58		cnfldex 20600	. . . . . 6 ⊢ ℂ_fld ∈ V
59		ovex 7308	. . . . . 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 19735	. . . . . 6 ⊢ ((ℂ_fld ∈ V ∧ (0[,]1) ∈ V) → ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1))))
63	58, 59, 62	mp2an 689	. . . . 5 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1)))
64	60	dfii4 24047	. . . . 5 ⊢ II = (TopOpen‘(ℂ_fld ↾_s (0[,]1)))
65	63, 64	mgptopn 19732	. . . 4 ⊢ II = (TopOpen‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
66		cnfldbas 20601	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
67	61, 66	mgpbas 19726	. . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
68		cnfldmul 20603	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
69	61, 68	mgpplusg 19724	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
70	7, 8	ax-mp 5	. . . . . 6 ⊢ · Fn (ℂ × ℂ)
71	67, 56, 69, 70, 4	ressplusf 31235	. . . . 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 23234	. . 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 31876	1 ⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 ifcif 4459 ↦ cmpt 5157 × cxp 5587 ^◡ccnv 5588 ↾ cres 5591 Fn wfn 6428 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 · cmul 10876 +∞cpnf 11006 ℝ^*cxr 11008 ≤ cle 11010 -cneg 11206 +_𝑒 cxad 12846 [,]cicc 13082 expce 15771 ↾_s cress 16941 ↾_t crest 17131 ordTopcordt 17210 +_𝑓cplusf 18323 mulGrpcmgp 19720 ℂ_fldccnfld 20597 TopOnctopon 22059 Cn ccn 22375 ×_t ctx 22711 TopMndctmd 23221 IIcii 24038 logclog 25710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-ordt 17212 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-ps 18284 df-tsr 18285 df-plusf 18325 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-subrg 20022 df-abv 20077 df-lmod 20125 df-scaf 20126 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-tmd 23223 df-tgp 23224 df-trg 23311 df-xms 23473 df-ms 23474 df-tms 23475 df-nm 23738 df-ngp 23739 df-nrg 23741 df-nlm 23742 df-ii 24040 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712

This theorem is referenced by: xrge0tmdALT 31896

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