xrge0pluscn - Mathbox for Thierry Arnoux

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Theorem xrge0pluscn 34131

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 34127	. 2 ⊢ 𝐹 ∈ (IIHomeo𝐽)
4		unitsscn 13451	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
5		xpss12 5640	. . . . 5 ⊢ (((0[,]1) ⊆ ℂ ∧ (0[,]1) ⊆ ℂ) → ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ))
6	4, 4, 5	mp2an 698	. . . 4 ⊢ ((0[,]1) × (0[,]1)) ⊆ (ℂ × ℂ)
7		ax-mulf 11116	. . . . 5 ⊢ · :(ℂ × ℂ)⟶ℂ
8		ffn 6662	. . . . 5 ⊢ ( · :(ℂ × ℂ)⟶ℂ → · Fn (ℂ × ℂ))
9		fnssresb 6614	. . . . 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 232	. . 3 ⊢ ( · ↾ ((0[,]1) × (0[,]1))) Fn ((0[,]1) × (0[,]1))
12		ovres 7529	. . . . 5 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) = (𝑢 · 𝑣))
13		iimulcl 24929	. . . . 5 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢 · 𝑣) ∈ (0[,]1))
14	12, 13	eqeltrd 2840	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1))
15	14	rgen2 3180	. . 3 ⊢ ∀𝑢 ∈ (0[,]1)∀𝑣 ∈ (0[,]1)(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1)
16		ffnov 7489	. . 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 717	. 2 ⊢ ( · ↾ ((0[,]1) × (0[,]1))):((0[,]1) × (0[,]1))⟶(0[,]1)
18		iccssxr 13381	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
19		xpss12 5640	. . . . . 6 ⊢ (((0[,]+∞) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^) → ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^ × ℝ^*))
20	18, 18, 19	mp2an 698	. . . . 5 ⊢ ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^* × ℝ^*)
21		xaddf 13174	. . . . . 6 ⊢ +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^
22		ffn 6662	. . . . . 6 ⊢ ( +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^ → +_𝑒 Fn (ℝ^* × ℝ^*))
23		fnssresb 6614	. . . . . 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 232	. . . 4 ⊢ ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞))
26		xrge0pluscn.1	. . . . 5 ⊢ + = ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))
27	26	fneq1i 6589	. . . 4 ⊢ ( + Fn ((0[,]+∞) × (0[,]+∞)) ↔ ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞)))
28	25, 27	mpbir 232	. . 3 ⊢ + Fn ((0[,]+∞) × (0[,]+∞))
29	26	oveqi 7376	. . . . 5 ⊢ (𝑎 + 𝑏) = (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏)
30		ovres 7529	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏) = (𝑎 +_𝑒 𝑏))
31		ge0xaddcl 13413	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
32	30, 31	eqeltrd 2840	. . . . 5 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏) ∈ (0[,]+∞))
33	29, 32	eqeltrid 2844	. . . 4 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 + 𝑏) ∈ (0[,]+∞))
34	33	rgen2 3180	. . 3 ⊢ ∀𝑎 ∈ (0[,]+∞)∀𝑏 ∈ (0[,]+∞)(𝑎 + 𝑏) ∈ (0[,]+∞)
35		ffnov 7489	. . 3 ⊢ ( + :((0[,]+∞) × (0[,]+∞))⟶(0[,]+∞) ↔ ( + Fn ((0[,]+∞) × (0[,]+∞)) ∧ ∀𝑎 ∈ (0[,]+∞)∀𝑏 ∈ (0[,]+∞)(𝑎 + 𝑏) ∈ (0[,]+∞)))
36	28, 34, 35	mpbir2an 717	. 2 ⊢ + :((0[,]+∞) × (0[,]+∞))⟶(0[,]+∞)
37		iitopon 24871	. 2 ⊢ II ∈ (TopOn‘(0[,]1))
38		letopon 23195	. . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
39		resttopon 23151	. . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ (0[,]+∞) ⊆ ℝ^) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)))
40	38, 18, 39	mp2an 698	. . 3 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))
41	2, 40	eqeltri 2836	. 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
42	26	oveqi 7376	. . . 4 ⊢ ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣))
43	1	xrge0iifcnv 34124	. . . . . . . 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 6774	. . . . . . 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 7031	. . . . 5 ⊢ (𝑢 ∈ (0[,]1) → (𝐹‘𝑢) ∈ (0[,]+∞))
48	46	ffvelcdmi 7031	. . . . 5 ⊢ (𝑣 ∈ (0[,]1) → (𝐹‘𝑣) ∈ (0[,]+∞))
49		ovres 7529	. . . . 5 ⊢ (((𝐹‘𝑢) ∈ (0[,]+∞) ∧ (𝐹‘𝑣) ∈ (0[,]+∞)) → ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
50	47, 48, 49	syl2an 602	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
51	42, 50	eqtrid 2787	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
52	1, 2	xrge0iifhom 34128	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘(𝑢 · 𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
53	12	eqcomd 2746	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢 · 𝑣) = (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣))
54	53	fveq2d 6838	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘(𝑢 · 𝑣)) = (𝐹‘(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣)))
55	51, 52, 54	3eqtr2rd 2782	. 2 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝐹‘(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣)) = ((𝐹‘𝑢) + (𝐹‘𝑣)))
56		eqid 2740	. . . 4 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = ((mulGrp‘ℂ_fld) ↾_s (0[,]1))
57	56	iistmd 34093	. . 3 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ TopMnd
58		cnfldex 21357	. . . . . 6 ⊢ ℂ_fld ∈ V
59		ovex 7396	. . . . . 6 ⊢ (0[,]1) ∈ V
60		eqid 2740	. . . . . . 7 ⊢ (ℂ_fld ↾_s (0[,]1)) = (ℂ_fld ↾_s (0[,]1))
61		eqid 2740	. . . . . . 7 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
62	60, 61	mgpress 20129	. . . . . 6 ⊢ ((ℂ_fld ∈ V ∧ (0[,]1) ∈ V) → ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1))))
63	58, 59, 62	mp2an 698	. . . . 5 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) = (mulGrp‘(ℂ_fld ↾_s (0[,]1)))
64	60	dfii4 24876	. . . . 5 ⊢ II = (TopOpen‘(ℂ_fld ↾_s (0[,]1)))
65	63, 64	mgptopn 20127	. . . 4 ⊢ II = (TopOpen‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
66		cnfldbas 21358	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
67	61, 66	mgpbas 20124	. . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
68		cnfldmul 21362	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
69	61, 68	mgpplusg 20123	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
70	7, 8	ax-mp 5	. . . . . 6 ⊢ · Fn (ℂ × ℂ)
71	67, 56, 69, 70, 4	ressplusf 33049	. . . . 5 ⊢ (+_𝑓‘((mulGrp‘ℂ_fld) ↾_s (0[,]1))) = ( · ↾ ((0[,]1) × (0[,]1)))
72	71	eqcomi 2749	. . . 4 ⊢ ( · ↾ ((0[,]1) × (0[,]1))) = (+_𝑓‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
73	65, 72	tmdcn 24073	. . 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 34117	1 ⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3054 Vcvv 3432 ⊆ wss 3890 ifcif 4461 ↦ cmpt 5160 × cxp 5623 ^◡ccnv 5624 ↾ cres 5627 Fn wfn 6487 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 0cc0 11036 1c1 11037 · cmul 11041 +∞cpnf 11174 ℝ^*cxr 11176 ≤ cle 11178 -cneg 11376 +_𝑒 cxad 13059 [,]cicc 13299 expce 16024 ↾_s cress 17198 ↾_t crest 17381 ordTopcordt 17461 +_𝑓cplusf 18603 mulGrpcmgp 20119 ℂ_fldccnfld 21354 TopOnctopon 22900 Cn ccn 23214 ×_t ctx 23550 TopMndctmd 24060 IIcii 24867 logclog 26543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115 ax-mulf 11116

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-ef 16030 df-sin 16032 df-cos 16033 df-pi 16035 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-ordt 17463 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-ps 18530 df-tsr 18531 df-plusf 18605 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-mulg 19042 df-subg 19097 df-cntz 19290 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-subrng 20525 df-subrg 20549 df-abv 20788 df-lmod 20859 df-scaf 20860 df-sra 21170 df-rgmod 21171 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-tmd 24062 df-tgp 24063 df-trg 24150 df-xms 24310 df-ms 24311 df-tms 24312 df-nm 24572 df-ngp 24573 df-nrg 24575 df-nlm 24576 df-ii 24869 df-cncf 24870 df-limc 25858 df-dv 25859 df-log 26545

This theorem is referenced by: xrge0tmdALT 34137

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