esumpr2 - Mathbox for Thierry Arnoux

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Theorem esumpr2 31436

Description: Extended sum over a pair, with a relaxed condition compared to esumpr 31435. (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 488	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
2		dfsn2 4538	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
3		preq2 4630	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
4	2, 3	syl5req 2846	. . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
5		esumeq1 31403	. . . . 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 31434	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
11	10	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
12	6, 11	eqtrd 2833	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = 𝐷)
13		esumpr2.1	. . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞)))
14		oveq2 7143	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 0))
15		0xr 10677	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
16		eleq1 2877	. . . . . . . . 9 ⊢ (𝐷 = 0 → (𝐷 ∈ ℝ^* ↔ 0 ∈ ℝ^*))
17	15, 16	mpbiri 261	. . . . . . . 8 ⊢ (𝐷 = 0 → 𝐷 ∈ ℝ^*)
18		xaddid1 12622	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ^* → (𝐷 +_𝑒 0) = 𝐷)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 0) = 𝐷)
20	14, 19	eqtrd 2833	. . . . . 6 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = 𝐷)
21		pnfxr 10684	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
22		eleq1 2877	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ∈ ℝ^* ↔ +∞ ∈ ℝ^*))
23	21, 22	mpbiri 261	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ∈ ℝ^*)
24		pnfnemnf 10685	. . . . . . . . 9 ⊢ +∞ ≠ -∞
25		neeq1 3049	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ≠ -∞ ↔ +∞ ≠ -∞))
26	24, 25	mpbiri 261	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ≠ -∞)
27		xaddpnf1 12607	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≠ -∞) → (𝐷 +_𝑒 +∞) = +∞)
28	23, 26, 27	syl2anc 587	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 +∞) = +∞)
29		oveq2 7143	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 +∞))
30		id 22	. . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞)
31	28, 29, 30	3eqtr4d 2843	. . . . . 6 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = 𝐷)
32	20, 31	jaoi 854	. . . . 5 ⊢ ((𝐷 = 0 ∨ 𝐷 = +∞) → (𝐷 +_𝑒 𝐷) = 𝐷)
33	13, 32	syl6 35	. . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 +_𝑒 𝐷) = 𝐷))
34	33	imp 410	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +_𝑒 𝐷) = 𝐷)
35		simpll 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝜑)
36		eqeq2 2810	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐴 ↔ 𝑘 = 𝐵))
37	36	biimprd 251	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐵 → 𝑘 = 𝐴))
38	1, 37	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑘 = 𝐵 → 𝑘 = 𝐴))
39	38	imp 410	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝑘 = 𝐴)
40	35, 39, 7	syl2anc 587	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷)
41		esumpr.4	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
42	41	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊)
43	9	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞))
44	40, 42, 43	esumsn 31434	. . . . 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 31434	. . . . . 6 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
48	47	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
49	44, 48	eqtr3d 2835	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸)
50	49	oveq2d 7151	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 𝐸))
51	12, 34, 50	3eqtr2d 2839	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
52	7	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
53	45	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
54	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑉)
55	41	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑊)
56	9	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ (0[,]+∞))
57	46	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐸 ∈ (0[,]+∞))
58		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
59	52, 53, 54, 55, 56, 57, 58	esumpr 31435	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
60	51, 59	pm2.61dane 3074	1 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {csn 4525 {cpr 4527 (class class class)co 7135 0cc0 10526 +∞cpnf 10661 -∞cmnf 10662 ℝ^*cxr 10663 +_𝑒 cxad 12493 [,]cicc 12729 Σ^*cesum 31396

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-ordt 16766 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-ps 17802 df-tsr 17803 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 df-abv 19581 df-lmod 19629 df-scaf 19630 df-sra 19937 df-rgmod 19938 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-tmd 22677 df-tgp 22678 df-tsms 22732 df-trg 22765 df-xms 22927 df-ms 22928 df-tms 22929 df-nm 23189 df-ngp 23190 df-nrg 23192 df-nlm 23193 df-ii 23482 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 df-esum 31397

This theorem is referenced by: measxun2 31579 measssd 31584 carsgclctun 31689

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