esumpr2 - Mathbox for Thierry Arnoux

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Theorem esumpr2 34173

Description: Extended sum over a pair, with a relaxed condition compared to esumpr 34172. (Contributed by Thierry Arnoux, 2-Jan-2017.)

**Hypotheses**
Ref	Expression
esumpr.1	⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
esumpr.2	⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
esumpr.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esumpr.4	⊢ (𝜑 → 𝐵 ∈ 𝑊)
esumpr.5	⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))
esumpr.6	⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))
esumpr2.1	⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞)))

**Assertion**
Ref	Expression
esumpr2	⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝐷,𝑘 𝑘,𝐸 𝜑,𝑘 𝑘,𝑉 𝑘,𝑊 Allowed substitution hint: 𝐶(𝑘)

**Proof of Theorem esumpr2**
Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
2		dfsn2 4591	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
3		preq2 4689	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
4	2, 3	eqtr2id 2782	. . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
5		esumeq1 34140	. . . . 5 ⊢ ({𝐴, 𝐵} = {𝐴} → Σ^𝑘 ∈ {𝐴, 𝐵}𝐶 = Σ^𝑘 ∈ {𝐴}𝐶)
6	1, 4, 5	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^𝑘 ∈ {𝐴, 𝐵}𝐶 = Σ^𝑘 ∈ {𝐴}𝐶)
7		esumpr.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
8		esumpr.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9		esumpr.5	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))
10	7, 8, 9	esumsn 34171	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
12	6, 11	eqtrd 2769	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = 𝐷)
13		esumpr2.1	. . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞)))
14		oveq2 7364	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 0))
15		0xr 11177	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
16		eleq1 2822	. . . . . . . . 9 ⊢ (𝐷 = 0 → (𝐷 ∈ ℝ^* ↔ 0 ∈ ℝ^*))
17	15, 16	mpbiri 258	. . . . . . . 8 ⊢ (𝐷 = 0 → 𝐷 ∈ ℝ^*)
18		xaddrid 13154	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ^* → (𝐷 +_𝑒 0) = 𝐷)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 0) = 𝐷)
20	14, 19	eqtrd 2769	. . . . . 6 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = 𝐷)
21		pnfxr 11184	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
22		eleq1 2822	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ∈ ℝ^* ↔ +∞ ∈ ℝ^*))
23	21, 22	mpbiri 258	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ∈ ℝ^*)
24		pnfnemnf 11185	. . . . . . . . 9 ⊢ +∞ ≠ -∞
25		neeq1 2992	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ≠ -∞ ↔ +∞ ≠ -∞))
26	24, 25	mpbiri 258	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ≠ -∞)
27		xaddpnf1 13139	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≠ -∞) → (𝐷 +_𝑒 +∞) = +∞)
28	23, 26, 27	syl2anc 584	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 +∞) = +∞)
29		oveq2 7364	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 +∞))
30		id 22	. . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞)
31	28, 29, 30	3eqtr4d 2779	. . . . . 6 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = 𝐷)
32	20, 31	jaoi 857	. . . . 5 ⊢ ((𝐷 = 0 ∨ 𝐷 = +∞) → (𝐷 +_𝑒 𝐷) = 𝐷)
33	13, 32	syl6 35	. . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 +_𝑒 𝐷) = 𝐷))
34	33	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +_𝑒 𝐷) = 𝐷)
35		simpll 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝜑)
36		eqeq2 2746	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐴 ↔ 𝑘 = 𝐵))
37	36	biimprd 248	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐵 → 𝑘 = 𝐴))
38	1, 37	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑘 = 𝐵 → 𝑘 = 𝐴))
39	38	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝑘 = 𝐴)
40	35, 39, 7	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷)
41		esumpr.4	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
42	41	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊)
43	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞))
44	40, 42, 43	esumsn 34171	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐷)
45		esumpr.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
46		esumpr.6	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))
47	45, 41, 46	esumsn 34171	. . . . . 6 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
48	47	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
49	44, 48	eqtr3d 2771	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸)
50	49	oveq2d 7372	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 𝐸))
51	12, 34, 50	3eqtr2d 2775	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
52	7	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
53	45	adantlr 715	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
54	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑉)
55	41	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑊)
56	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ (0[,]+∞))
57	46	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐸 ∈ (0[,]+∞))
58		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
59	52, 53, 54, 55, 56, 57, 58	esumpr 34172	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
60	51, 59	pm2.61dane 3017	1 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 {csn 4578 {cpr 4580 (class class class)co 7356 0cc0 11024 +∞cpnf 11161 -∞cmnf 11162 ℝ^*cxr 11163 +_𝑒 cxad 13022 [,]cicc 13262 Σ^*cesum 34133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103 ax-mulf 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-fac 14195 df-bc 14224 df-hash 14252 df-shft 14988 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-ef 15988 df-sin 15990 df-cos 15991 df-pi 15993 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-ordt 17420 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-ps 18487 df-tsr 18488 df-plusf 18562 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-subg 19051 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-subrng 20477 df-subrg 20501 df-abv 20740 df-lmod 20811 df-scaf 20812 df-sra 21123 df-rgmod 21124 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-tmd 24014 df-tgp 24015 df-tsms 24069 df-trg 24102 df-xms 24262 df-ms 24263 df-tms 24264 df-nm 24524 df-ngp 24525 df-nrg 24527 df-nlm 24528 df-ii 24824 df-cncf 24825 df-limc 25821 df-dv 25822 df-log 26519 df-esum 34134

This theorem is referenced by: measxun2 34316 measssd 34321 carsgclctun 34427

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