xrge0pluscn - Mathbox for Thierry Arnoux

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Theorem xrge0pluscn 31604

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 31600	. 2 ⊢ 𝐹 ∈ (IIHomeo𝐽)
4		unitsscn 13088	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
5		xpss12 5566	. . . . 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 10809	. . . . 5 ⊢ · :(ℂ × ℂ)⟶ℂ
8		ffn 6545	. . . . 5 ⊢ ( · :(ℂ × ℂ)⟶ℂ → · Fn (ℂ × ℂ))
9		fnssresb 6499	. . . . 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 234	. . 3 ⊢ ( · ↾ ((0[,]1) × (0[,]1))) Fn ((0[,]1) × (0[,]1))
12		ovres 7374	. . . . 5 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) = (𝑢 · 𝑣))
13		iimulcl 23834	. . . . 5 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢 · 𝑣) ∈ (0[,]1))
14	12, 13	eqeltrd 2838	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → (𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1))
15	14	rgen2 3124	. . 3 ⊢ ∀𝑢 ∈ (0[,]1)∀𝑣 ∈ (0[,]1)(𝑢( · ↾ ((0[,]1) × (0[,]1)))𝑣) ∈ (0[,]1)
16		ffnov 7337	. . 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 13018	. . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ^*
19		xpss12 5566	. . . . . 6 ⊢ (((0[,]+∞) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^) → ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^ × ℝ^*))
20	18, 18, 19	mp2an 692	. . . . 5 ⊢ ((0[,]+∞) × (0[,]+∞)) ⊆ (ℝ^* × ℝ^*)
21		xaddf 12814	. . . . . 6 ⊢ +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^
22		ffn 6545	. . . . . 6 ⊢ ( +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^ → +_𝑒 Fn (ℝ^* × ℝ^*))
23		fnssresb 6499	. . . . . 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 234	. . . 4 ⊢ ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞))
26		xrge0pluscn.1	. . . . 5 ⊢ + = ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))
27	26	fneq1i 6476	. . . 4 ⊢ ( + Fn ((0[,]+∞) × (0[,]+∞)) ↔ ( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞))) Fn ((0[,]+∞) × (0[,]+∞)))
28	25, 27	mpbir 234	. . 3 ⊢ + Fn ((0[,]+∞) × (0[,]+∞))
29	26	oveqi 7226	. . . . 5 ⊢ (𝑎 + 𝑏) = (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏)
30		ovres 7374	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏) = (𝑎 +_𝑒 𝑏))
31		ge0xaddcl 13050	. . . . . 6 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 +_𝑒 𝑏) ∈ (0[,]+∞))
32	30, 31	eqeltrd 2838	. . . . 5 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))𝑏) ∈ (0[,]+∞))
33	29, 32	eqeltrid 2842	. . . 4 ⊢ ((𝑎 ∈ (0[,]+∞) ∧ 𝑏 ∈ (0[,]+∞)) → (𝑎 + 𝑏) ∈ (0[,]+∞))
34	33	rgen2 3124	. . 3 ⊢ ∀𝑎 ∈ (0[,]+∞)∀𝑏 ∈ (0[,]+∞)(𝑎 + 𝑏) ∈ (0[,]+∞)
35		ffnov 7337	. . 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 23776	. 2 ⊢ II ∈ (TopOn‘(0[,]1))
38		letopon 22102	. . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
39		resttopon 22058	. . . 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 2834	. 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
42	26	oveqi 7226	. . . 4 ⊢ ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣))
43	1	xrge0iifcnv 31597	. . . . . . . 8 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))
44	43	simpli 487	. . . . . . 7 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
45		f1of 6661	. . . . . . 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 6903	. . . . 5 ⊢ (𝑢 ∈ (0[,]1) → (𝐹‘𝑢) ∈ (0[,]+∞))
48	46	ffvelrni 6903	. . . . 5 ⊢ (𝑣 ∈ (0[,]1) → (𝐹‘𝑣) ∈ (0[,]+∞))
49		ovres 7374	. . . . 5 ⊢ (((𝐹‘𝑢) ∈ (0[,]+∞) ∧ (𝐹‘𝑣) ∈ (0[,]+∞)) → ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
50	47, 48, 49	syl2an 599	. . . 4 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘𝑢)( +_𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))(𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
51	42, 50	syl5eq 2790	. . 3 ⊢ ((𝑢 ∈ (0[,]1) ∧ 𝑣 ∈ (0[,]1)) → ((𝐹‘𝑢) + (𝐹‘𝑣)) = ((𝐹‘𝑢) +_𝑒 (𝐹‘𝑣)))
52	1, 2	xrge0iifhom 31601	. . 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 6721	. . 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 31566	. . 3 ⊢ ((mulGrp‘ℂ_fld) ↾_s (0[,]1)) ∈ TopMnd
58		cnfldex 20366	. . . . . 6 ⊢ ℂ_fld ∈ V
59		ovex 7246	. . . . . 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 19515	. . . . . 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 23781	. . . . 5 ⊢ II = (TopOpen‘(ℂ_fld ↾_s (0[,]1)))
65	63, 64	mgptopn 19513	. . . 4 ⊢ II = (TopOpen‘((mulGrp‘ℂ_fld) ↾_s (0[,]1)))
66		cnfldbas 20367	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
67	61, 66	mgpbas 19510	. . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
68		cnfldmul 20369	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
69	61, 68	mgpplusg 19508	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
70	7, 8	ax-mp 5	. . . . . 6 ⊢ · Fn (ℂ × ℂ)
71	67, 56, 69, 70, 4	ressplusf 30955	. . . . 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 22980	. . 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 31590	1 ⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ⊆ wss 3866 ifcif 4439 ↦ cmpt 5135 × cxp 5549 ^◡ccnv 5550 ↾ cres 5553 Fn wfn 6375 ⟶wf 6376 –1-1-onto→wf1o 6379 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 0cc0 10729 1c1 10730 · cmul 10734 +∞cpnf 10864 ℝ^*cxr 10866 ≤ cle 10868 -cneg 11063 +_𝑒 cxad 12702 [,]cicc 12938 expce 15623 ↾_s cress 16784 ↾_t crest 16925 ordTopcordt 17004 +_𝑓cplusf 18111 mulGrpcmgp 19504 ℂ_fldccnfld 20363 TopOnctopon 21807 Cn ccn 22121 ×_t ctx 22457 TopMndctmd 22967 IIcii 23772 logclog 25443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-ef 15629 df-sin 15631 df-cos 15632 df-pi 15634 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-ordt 17006 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-ps 18072 df-tsr 18073 df-plusf 18113 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-subrg 19798 df-abv 19853 df-lmod 19901 df-scaf 19902 df-sra 20209 df-rgmod 20210 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-tmd 22969 df-tgp 22970 df-trg 23057 df-xms 23218 df-ms 23219 df-tms 23220 df-nm 23480 df-ngp 23481 df-nrg 23483 df-nlm 23484 df-ii 23774 df-cncf 23775 df-limc 24763 df-dv 24764 df-log 25445

This theorem is referenced by: xrge0tmdALT 31610

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