esumpr2 - Mathbox for Thierry Arnoux

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Theorem esumpr2 30727

Description: Extended sum over a pair, with a relaxed condition compared to esumpr 30726. (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 479	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
2		dfsn2 4411	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
3		preq2 4501	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
4	2, 3	syl5req 2827	. . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
5		esumeq1 30694	. . . . 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 30725	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
11	10	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴}𝐶 = 𝐷)
12	6, 11	eqtrd 2814	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = 𝐷)
13		esumpr2.1	. . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞)))
14		oveq2 6930	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 0))
15		0xr 10423	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
16		eleq1 2847	. . . . . . . . 9 ⊢ (𝐷 = 0 → (𝐷 ∈ ℝ^* ↔ 0 ∈ ℝ^*))
17	15, 16	mpbiri 250	. . . . . . . 8 ⊢ (𝐷 = 0 → 𝐷 ∈ ℝ^*)
18		xaddid1 12384	. . . . . . . 8 ⊢ (𝐷 ∈ ℝ^* → (𝐷 +_𝑒 0) = 𝐷)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 0) = 𝐷)
20	14, 19	eqtrd 2814	. . . . . 6 ⊢ (𝐷 = 0 → (𝐷 +_𝑒 𝐷) = 𝐷)
21		pnfxr 10430	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
22		eleq1 2847	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ∈ ℝ^* ↔ +∞ ∈ ℝ^*))
23	21, 22	mpbiri 250	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ∈ ℝ^*)
24		pnfnemnf 10432	. . . . . . . . 9 ⊢ +∞ ≠ -∞
25		neeq1 3031	. . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ≠ -∞ ↔ +∞ ≠ -∞))
26	24, 25	mpbiri 250	. . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ≠ -∞)
27		xaddpnf1 12369	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≠ -∞) → (𝐷 +_𝑒 +∞) = +∞)
28	23, 26, 27	syl2anc 579	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 +∞) = +∞)
29		oveq2 6930	. . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 +∞))
30		id 22	. . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞)
31	28, 29, 30	3eqtr4d 2824	. . . . . 6 ⊢ (𝐷 = +∞ → (𝐷 +_𝑒 𝐷) = 𝐷)
32	20, 31	jaoi 846	. . . . 5 ⊢ ((𝐷 = 0 ∨ 𝐷 = +∞) → (𝐷 +_𝑒 𝐷) = 𝐷)
33	13, 32	syl6 35	. . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 +_𝑒 𝐷) = 𝐷))
34	33	imp 397	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +_𝑒 𝐷) = 𝐷)
35		simpll 757	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝜑)
36		eqeq2 2789	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐴 ↔ 𝑘 = 𝐵))
37	36	biimprd 240	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐵 → 𝑘 = 𝐴))
38	1, 37	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑘 = 𝐵 → 𝑘 = 𝐴))
39	38	imp 397	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝑘 = 𝐴)
40	35, 39, 7	syl2anc 579	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷)
41		esumpr.4	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
42	41	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊)
43	9	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞))
44	40, 42, 43	esumsn 30725	. . . . 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 30725	. . . . . 6 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
48	47	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐵}𝐶 = 𝐸)
49	44, 48	eqtr3d 2816	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸)
50	49	oveq2d 6938	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +_𝑒 𝐷) = (𝐷 +_𝑒 𝐸))
51	12, 34, 50	3eqtr2d 2820	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
52	7	adantlr 705	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
53	45	adantlr 705	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)
54	8	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑉)
55	41	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑊)
56	9	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ (0[,]+∞))
57	46	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐸 ∈ (0[,]+∞))
58		simpr 479	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
59	52, 53, 54, 55, 56, 57, 58	esumpr 30726	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))
60	51, 59	pm2.61dane 3057	1 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 {csn 4398 {cpr 4400 (class class class)co 6922 0cc0 10272 +∞cpnf 10408 -∞cmnf 10409 ℝ^*cxr 10410 +_𝑒 cxad 12255 [,]cicc 12490 Σ^*cesum 30687

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-ef 15200 df-sin 15202 df-cos 15203 df-pi 15205 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-ordt 16547 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-ps 17586 df-tsr 17587 df-plusf 17627 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-subrg 19170 df-abv 19209 df-lmod 19257 df-scaf 19258 df-sra 19569 df-rgmod 19570 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-tmd 22284 df-tgp 22285 df-tsms 22338 df-trg 22371 df-xms 22533 df-ms 22534 df-tms 22535 df-nm 22795 df-ngp 22796 df-nrg 22798 df-nlm 22799 df-ii 23088 df-cncf 23089 df-limc 24067 df-dv 24068 df-log 24740 df-esum 30688

This theorem is referenced by: measxun2 30871 measssd 30876 carsgclctun 30981

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