Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpr2 | Structured version Visualization version GIF version |
Description: Extended sum over a pair, with a relaxed condition compared to esumpr 32330. (Contributed by Thierry Arnoux, 2-Jan-2017.) |
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 = +∞))) |
Ref | Expression |
---|---|
esumpr2 | ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
2 | dfsn2 4591 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | preq2 4687 | . . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
4 | 2, 3 | eqtr2id 2790 | . . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | esumeq1 32298 | . . . . 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 32329 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴}𝐶 = 𝐷) |
11 | 10 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴}𝐶 = 𝐷) |
12 | 6, 11 | eqtrd 2777 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = 𝐷) |
13 | esumpr2.1 | . . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞))) | |
14 | oveq2 7350 | . . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 0)) | |
15 | 0xr 11128 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
16 | eleq1 2825 | . . . . . . . . 9 ⊢ (𝐷 = 0 → (𝐷 ∈ ℝ* ↔ 0 ∈ ℝ*)) | |
17 | 15, 16 | mpbiri 258 | . . . . . . . 8 ⊢ (𝐷 = 0 → 𝐷 ∈ ℝ*) |
18 | xaddid1 13081 | . . . . . . . 8 ⊢ (𝐷 ∈ ℝ* → (𝐷 +𝑒 0) = 𝐷) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +𝑒 0) = 𝐷) |
20 | 14, 19 | eqtrd 2777 | . . . . . 6 ⊢ (𝐷 = 0 → (𝐷 +𝑒 𝐷) = 𝐷) |
21 | pnfxr 11135 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
22 | eleq1 2825 | . . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ∈ ℝ* ↔ +∞ ∈ ℝ*)) | |
23 | 21, 22 | mpbiri 258 | . . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ∈ ℝ*) |
24 | pnfnemnf 11136 | . . . . . . . . 9 ⊢ +∞ ≠ -∞ | |
25 | neeq1 3004 | . . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ≠ -∞ ↔ +∞ ≠ -∞)) | |
26 | 24, 25 | mpbiri 258 | . . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ≠ -∞) |
27 | xaddpnf1 13066 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞) | |
28 | 23, 26, 27 | syl2anc 585 | . . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 +∞) = +∞) |
29 | oveq2 7350 | . . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 +∞)) | |
30 | id 22 | . . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞) | |
31 | 28, 29, 30 | 3eqtr4d 2787 | . . . . . 6 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐷) = 𝐷) |
32 | 20, 31 | jaoi 855 | . . . . 5 ⊢ ((𝐷 = 0 ∨ 𝐷 = +∞) → (𝐷 +𝑒 𝐷) = 𝐷) |
33 | 13, 32 | syl6 35 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 +𝑒 𝐷) = 𝐷)) |
34 | 33 | imp 408 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐷) = 𝐷) |
35 | simpll 765 | . . . . . . 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 408 | . . . . . . 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 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊) |
43 | 9 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
44 | 40, 42, 43 | esumsn 32329 | . . . . 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 32329 | . . . . . 6 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐸) |
48 | 47 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐸) |
49 | 44, 48 | eqtr3d 2779 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸) |
50 | 49 | oveq2d 7358 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 𝐸)) |
51 | 12, 34, 50 | 3eqtr2d 2783 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
52 | 7 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) |
53 | 45 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) |
54 | 8 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑉) |
55 | 41 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑊) |
56 | 9 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ (0[,]+∞)) |
57 | 46 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐸 ∈ (0[,]+∞)) |
58 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵) | |
59 | 52, 53, 54, 55, 56, 57, 58 | esumpr 32330 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
60 | 51, 59 | pm2.61dane 3030 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 {csn 4578 {cpr 4580 (class class class)co 7342 0cc0 10977 +∞cpnf 11112 -∞cmnf 11113 ℝ*cxr 11114 +𝑒 cxad 12952 [,]cicc 13188 Σ*cesum 32291 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5234 ax-sep 5248 ax-nul 5255 ax-pow 5313 ax-pr 5377 ax-un 7655 ax-inf2 9503 ax-cnex 11033 ax-resscn 11034 ax-1cn 11035 ax-icn 11036 ax-addcl 11037 ax-addrcl 11038 ax-mulcl 11039 ax-mulrcl 11040 ax-mulcom 11041 ax-addass 11042 ax-mulass 11043 ax-distr 11044 ax-i2m1 11045 ax-1ne0 11046 ax-1rid 11047 ax-rnegex 11048 ax-rrecex 11049 ax-cnre 11050 ax-pre-lttri 11051 ax-pre-lttrn 11052 ax-pre-ltadd 11053 ax-pre-mulgt0 11054 ax-pre-sup 11055 ax-addf 11056 ax-mulf 11057 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3732 df-csb 3848 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3921 df-nul 4275 df-if 4479 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4858 df-int 4900 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5181 df-tr 5215 df-id 5523 df-eprel 5529 df-po 5537 df-so 5538 df-fr 5580 df-se 5581 df-we 5582 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 6243 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6436 df-fun 6486 df-fn 6487 df-f 6488 df-f1 6489 df-fo 6490 df-f1o 6491 df-fv 6492 df-isom 6493 df-riota 7298 df-ov 7345 df-oprab 7346 df-mpo 7347 df-of 7600 df-om 7786 df-1st 7904 df-2nd 7905 df-supp 8053 df-frecs 8172 df-wrecs 8203 df-recs 8277 df-rdg 8316 df-1o 8372 df-2o 8373 df-er 8574 df-map 8693 df-pm 8694 df-ixp 8762 df-en 8810 df-dom 8811 df-sdom 8812 df-fin 8813 df-fsupp 9232 df-fi 9273 df-sup 9304 df-inf 9305 df-oi 9372 df-card 9801 df-pnf 11117 df-mnf 11118 df-xr 11119 df-ltxr 11120 df-le 11121 df-sub 11313 df-neg 11314 df-div 11739 df-nn 12080 df-2 12142 df-3 12143 df-4 12144 df-5 12145 df-6 12146 df-7 12147 df-8 12148 df-9 12149 df-n0 12340 df-z 12426 df-dec 12544 df-uz 12689 df-q 12795 df-rp 12837 df-xneg 12954 df-xadd 12955 df-xmul 12956 df-ioo 13189 df-ioc 13190 df-ico 13191 df-icc 13192 df-fz 13346 df-fzo 13489 df-fl 13618 df-mod 13696 df-seq 13828 df-exp 13889 df-fac 14094 df-bc 14123 df-hash 14151 df-shft 14878 df-cj 14910 df-re 14911 df-im 14912 df-sqrt 15046 df-abs 15047 df-limsup 15280 df-clim 15297 df-rlim 15298 df-sum 15498 df-ef 15877 df-sin 15879 df-cos 15880 df-pi 15882 df-struct 16946 df-sets 16963 df-slot 16981 df-ndx 16993 df-base 17011 df-ress 17040 df-plusg 17073 df-mulr 17074 df-starv 17075 df-sca 17076 df-vsca 17077 df-ip 17078 df-tset 17079 df-ple 17080 df-ds 17082 df-unif 17083 df-hom 17084 df-cco 17085 df-rest 17231 df-topn 17232 df-0g 17250 df-gsum 17251 df-topgen 17252 df-pt 17253 df-prds 17256 df-ordt 17310 df-xrs 17311 df-qtop 17316 df-imas 17317 df-xps 17319 df-mre 17393 df-mrc 17394 df-acs 17396 df-ps 18382 df-tsr 18383 df-plusf 18423 df-mgm 18424 df-sgrp 18473 df-mnd 18484 df-mhm 18528 df-submnd 18529 df-grp 18677 df-minusg 18678 df-sbg 18679 df-mulg 18798 df-subg 18849 df-cntz 19020 df-cmn 19484 df-abl 19485 df-mgp 19816 df-ur 19833 df-ring 19880 df-cring 19881 df-subrg 20127 df-abv 20183 df-lmod 20231 df-scaf 20232 df-sra 20540 df-rgmod 20541 df-psmet 20695 df-xmet 20696 df-met 20697 df-bl 20698 df-mopn 20699 df-fbas 20700 df-fg 20701 df-cnfld 20704 df-top 22149 df-topon 22166 df-topsp 22188 df-bases 22202 df-cld 22276 df-ntr 22277 df-cls 22278 df-nei 22355 df-lp 22393 df-perf 22394 df-cn 22484 df-cnp 22485 df-haus 22572 df-tx 22819 df-hmeo 23012 df-fil 23103 df-fm 23195 df-flim 23196 df-flf 23197 df-tmd 23329 df-tgp 23330 df-tsms 23384 df-trg 23417 df-xms 23579 df-ms 23580 df-tms 23581 df-nm 23844 df-ngp 23845 df-nrg 23847 df-nlm 23848 df-ii 24146 df-cncf 24147 df-limc 25136 df-dv 25137 df-log 25818 df-esum 32292 |
This theorem is referenced by: measxun2 32474 measssd 32479 carsgclctun 32586 |
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