esumpr2 - Mathbox for Thierry Arnoux

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Theorem esumpr2 34230

Description: Extended sum over a pair, with a relaxed condition compared to esumpr 34229. (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 4581	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
3		preq2 4679	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
4	2, 3	eqtr2id 2785	. . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
5		esumeq1 34197	. . . . 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 34228	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
12	6, 11	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = 𝐷)
13		esumpr2.1	. . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞)))
14		oveq2 7369	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 0))
15		0xr 11186	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
16		eleq1 2825	. . . . . . . . 9 ⊢ (𝐷 = 0 → (𝐷 ∈ ℝ^* ↔ 0 ∈ ℝ^*))
17	15, 16	mpbiri 258	. . . . . . . 8 ⊢ (𝐷 = 0 → 𝐷 ∈ ℝ^*)
18		xaddrid 13187	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ^* → (𝐷 +_𝑒 0) = 𝐷)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 0) = 𝐷)
20	14, 19	eqtrd 2772	. . . . . 6 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = 𝐷)
21		pnfxr 11193	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
22		eleq1 2825	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ∈ ℝ^* ↔ +∞ ∈ ℝ^*))
23	21, 22	mpbiri 258	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ∈ ℝ^*)
24		pnfnemnf 11194	. . . . . . . . 9 ⊢ +∞ ≠ -∞
25		neeq1 2995	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ≠ -∞ ↔ +∞ ≠ -∞))
26	24, 25	mpbiri 258	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ≠ -∞)
27		xaddpnf1 13172	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≠ -∞) → (𝐷 +_𝑒 +∞) = +∞)
28	23, 26, 27	syl2anc 585	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 +∞) = +∞)
29		oveq2 7369	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 +∞))
30		id 22	. . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞)
31	28, 29, 30	3eqtr4d 2782	. . . . . 6 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = 𝐷)
32	20, 31	jaoi 858	. . . . 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 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝜑)
36		eqeq2 2749	. . . . . . . . . 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 585	. . . . . 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 34228	. . . . 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 34228	. . . . . 6 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
48	47	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
49	44, 48	eqtr3d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸)
50	49	oveq2d 7377	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 𝐸))
51	12, 34, 50	3eqtr2d 2778	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
52	7	adantlr 716	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
53	45	adantlr 716	. . 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 34229	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
60	51, 59	pm2.61dane 3020	1 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 {csn 4568 {cpr 4570 (class class class)co 7361 0cc0 11032 +∞cpnf 11170 -∞cmnf 11171 ℝ^*cxr 11172 +_𝑒 cxad 13055 [,]cicc 13295 Σ^*cesum 34190

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111 ax-mulf 11112

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 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 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-sum 15643 df-ef 16026 df-sin 16028 df-cos 16029 df-pi 16031 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-ordt 17459 df-xrs 17460 df-qtop 17465 df-imas 17466 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-ps 18526 df-tsr 18527 df-plusf 18601 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-mhm 18745 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-mulg 19038 df-subg 19093 df-cntz 19286 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-cring 20211 df-subrng 20517 df-subrg 20541 df-abv 20780 df-lmod 20851 df-scaf 20852 df-sra 21163 df-rgmod 21164 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-haus 23293 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-tmd 24050 df-tgp 24051 df-tsms 24105 df-trg 24138 df-xms 24298 df-ms 24299 df-tms 24300 df-nm 24560 df-ngp 24561 df-nrg 24563 df-nlm 24564 df-ii 24857 df-cncf 24858 df-limc 25846 df-dv 25847 df-log 26536 df-esum 34191

This theorem is referenced by: measxun2 34373 measssd 34378 carsgclctun 34484

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