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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 34047. (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 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
2 | dfsn2 4644 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | preq2 4739 | . . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
4 | 2, 3 | eqtr2id 2788 | . . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | esumeq1 34015 | . . . . 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 34046 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴}𝐶 = 𝐷) |
11 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴}𝐶 = 𝐷) |
12 | 6, 11 | eqtrd 2775 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = 𝐷) |
13 | esumpr2.1 | . . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞))) | |
14 | oveq2 7439 | . . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 0)) | |
15 | 0xr 11306 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
16 | eleq1 2827 | . . . . . . . . 9 ⊢ (𝐷 = 0 → (𝐷 ∈ ℝ* ↔ 0 ∈ ℝ*)) | |
17 | 15, 16 | mpbiri 258 | . . . . . . . 8 ⊢ (𝐷 = 0 → 𝐷 ∈ ℝ*) |
18 | xaddrid 13280 | . . . . . . . 8 ⊢ (𝐷 ∈ ℝ* → (𝐷 +𝑒 0) = 𝐷) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +𝑒 0) = 𝐷) |
20 | 14, 19 | eqtrd 2775 | . . . . . 6 ⊢ (𝐷 = 0 → (𝐷 +𝑒 𝐷) = 𝐷) |
21 | pnfxr 11313 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
22 | eleq1 2827 | . . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ∈ ℝ* ↔ +∞ ∈ ℝ*)) | |
23 | 21, 22 | mpbiri 258 | . . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ∈ ℝ*) |
24 | pnfnemnf 11314 | . . . . . . . . 9 ⊢ +∞ ≠ -∞ | |
25 | neeq1 3001 | . . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ≠ -∞ ↔ +∞ ≠ -∞)) | |
26 | 24, 25 | mpbiri 258 | . . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ≠ -∞) |
27 | xaddpnf1 13265 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞) | |
28 | 23, 26, 27 | syl2anc 584 | . . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 +∞) = +∞) |
29 | oveq2 7439 | . . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 +∞)) | |
30 | id 22 | . . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞) | |
31 | 28, 29, 30 | 3eqtr4d 2785 | . . . . . 6 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐷) = 𝐷) |
32 | 20, 31 | jaoi 857 | . . . . 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 2747 | . . . . . . . . . 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 584 | . . . . . 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 34046 | . . . . 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 34046 | . . . . . 6 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐸) |
48 | 47 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐸) |
49 | 44, 48 | eqtr3d 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸) |
50 | 49 | oveq2d 7447 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 𝐸)) |
51 | 12, 34, 50 | 3eqtr2d 2781 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
52 | 7 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) |
53 | 45 | adantlr 715 | . . 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 34047 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
60 | 51, 59 | pm2.61dane 3027 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {csn 4631 {cpr 4633 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 -∞cmnf 11291 ℝ*cxr 11292 +𝑒 cxad 13150 [,]cicc 13387 Σ*cesum 34008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-ordt 17548 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-ps 18624 df-tsr 18625 df-plusf 18665 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-scaf 20878 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tmd 24096 df-tgp 24097 df-tsms 24151 df-trg 24184 df-xms 24346 df-ms 24347 df-tms 24348 df-nm 24611 df-ngp 24612 df-nrg 24614 df-nlm 24615 df-ii 24917 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-esum 34009 |
This theorem is referenced by: measxun2 34191 measssd 34196 carsgclctun 34303 |
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