esumpr2 - Mathbox for Thierry Arnoux

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Theorem esumpr2 29456

Description: Extended sum over a pair, with a relaxed condition compared to esumpr 29455. (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 476	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
2		dfsn2 4138	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
3		preq2 4213	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
4	2, 3	syl5req 2657	. . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
5		esumeq1 29423	. . . . 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 29454	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
12	6, 11	eqtrd 2644	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = 𝐷)
13		esumpr2.1	. . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞)))
14		oveq2 6557	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 0))
15		0xr 9965	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
16		eleq1 2676	. . . . . . . . 9 ⊢ (𝐷 = 0 → (𝐷 ∈ ℝ^* ↔ 0 ∈ ℝ^*))
17	15, 16	mpbiri 247	. . . . . . . 8 ⊢ (𝐷 = 0 → 𝐷 ∈ ℝ^*)
18		xaddid1 11946	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ^* → (𝐷 +_𝑒 0) = 𝐷)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 0) = 𝐷)
20	14, 19	eqtrd 2644	. . . . . 6 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = 𝐷)
21		pnfxr 9971	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
22		eleq1 2676	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ∈ ℝ^* ↔ +∞ ∈ ℝ^*))
23	21, 22	mpbiri 247	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ∈ ℝ^*)
24		pnfnemnf 9973	. . . . . . . . 9 ⊢ +∞ ≠ -∞
25		neeq1 2844	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ≠ -∞ ↔ +∞ ≠ -∞))
26	24, 25	mpbiri 247	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ≠ -∞)
27		xaddpnf1 11931	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≠ -∞) → (𝐷 +_𝑒 +∞) = +∞)
28	23, 26, 27	syl2anc 691	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 +∞) = +∞)
29		oveq2 6557	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 +∞))
30		id 22	. . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞)
31	28, 29, 30	3eqtr4d 2654	. . . . . 6 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = 𝐷)
32	20, 31	jaoi 393	. . . . 5 ⊢ ((𝐷 = 0 ∨ 𝐷 = +∞) → (𝐷 +_𝑒 𝐷) = 𝐷)
33	13, 32	syl6 34	. . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 +_𝑒 𝐷) = 𝐷))
34	33	imp 444	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +_𝑒 𝐷) = 𝐷)
35		simpll 786	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝜑)
36		eqeq2 2621	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐴 ↔ 𝑘 = 𝐵))
37	36	biimprd 237	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐵 → 𝑘 = 𝐴))
38	1, 37	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑘 = 𝐵 → 𝑘 = 𝐴))
39	38	imp 444	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝑘 = 𝐴)
40	35, 39, 7	syl2anc 691	. . . . . 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 29454	. . . . 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 29454	. . . . . 6 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
48	47	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
49	44, 48	eqtr3d 2646	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸)
50	49	oveq2d 6565	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 𝐸))
51	12, 34, 50	3eqtr2d 2650	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
52	7	adantlr 747	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
53	45	adantlr 747	. . 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 476	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
59	52, 53, 54, 55, 56, 57, 58	esumpr 29455	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
60	51, 59	pm2.61dane 2869	1 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {csn 4125 {cpr 4127 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 +_𝑒 cxad 11820 [,]cicc 12049 Σ^*cesum 29416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-ordt 15984 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-abv 18640 df-lmod 18688 df-scaf 18689 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-tsms 21740 df-trg 21773 df-xms 21935 df-ms 21936 df-tms 21937 df-nm 22197 df-ngp 22198 df-nrg 22200 df-nlm 22201 df-ii 22488 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-esum 29417

This theorem is referenced by: measxun2 29600 measssd 29605 carsgclctun 29710

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