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Mirrors > Home > MPE Home > Th. List > sumsnf | Structured version Visualization version GIF version |
Description: A sum of a singleton is the term. A version of sumsn 15095 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
sumsnf.1 | ⊢ Ⅎ𝑘𝐵 |
sumsnf.2 | ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
sumsnf | ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑚𝐴 | |
2 | nfcsb1v 3906 | . . . 4 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐴 | |
3 | csbeq1a 3896 | . . . 4 ⊢ (𝑘 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑘⦌𝐴) | |
4 | 1, 2, 3 | cbvsumi 15048 | . . 3 ⊢ Σ𝑘 ∈ {𝑀}𝐴 = Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 |
5 | csbeq1 3885 | . . . 4 ⊢ (𝑚 = ({〈1, 𝑀〉}‘𝑛) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴) | |
6 | 1nn 11643 | . . . . 5 ⊢ 1 ∈ ℕ | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → 1 ∈ ℕ) |
8 | simpl 485 | . . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → 𝑀 ∈ 𝑉) | |
9 | f1osng 6649 | . . . . . 6 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀}) | |
10 | 6, 8, 9 | sylancr 589 | . . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀}) |
11 | 1z 12006 | . . . . . 6 ⊢ 1 ∈ ℤ | |
12 | fzsn 12943 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
13 | f1oeq2 6599 | . . . . . 6 ⊢ ((1...1) = {1} → ({〈1, 𝑀〉}:(1...1)–1-1-onto→{𝑀} ↔ {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀})) | |
14 | 11, 12, 13 | mp2b 10 | . . . . 5 ⊢ ({〈1, 𝑀〉}:(1...1)–1-1-onto→{𝑀} ↔ {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀}) |
15 | 10, 14 | sylibr 236 | . . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → {〈1, 𝑀〉}:(1...1)–1-1-onto→{𝑀}) |
16 | elsni 4577 | . . . . . . 7 ⊢ (𝑚 ∈ {𝑀} → 𝑚 = 𝑀) | |
17 | 16 | adantl 484 | . . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → 𝑚 = 𝑀) |
18 | 17 | csbeq1d 3886 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
19 | sumsnf.1 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐵 | |
20 | 19 | a1i 11 | . . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → Ⅎ𝑘𝐵) |
21 | sumsnf.2 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) | |
22 | 20, 21 | csbiegf 3915 | . . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
23 | 22 | ad2antrr 724 | . . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
24 | simplr 767 | . . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → 𝐵 ∈ ℂ) | |
25 | 23, 24 | eqeltrd 2913 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) |
26 | 18, 25 | eqeltrd 2913 | . . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → ⦋𝑚 / 𝑘⦌𝐴 ∈ ℂ) |
27 | 22 | ad2antrr 724 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
28 | elfz1eq 12912 | . . . . . . . 8 ⊢ (𝑛 ∈ (1...1) → 𝑛 = 1) | |
29 | 28 | fveq2d 6668 | . . . . . . 7 ⊢ (𝑛 ∈ (1...1) → ({〈1, 𝑀〉}‘𝑛) = ({〈1, 𝑀〉}‘1)) |
30 | fvsng 6936 | . . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → ({〈1, 𝑀〉}‘1) = 𝑀) | |
31 | 6, 8, 30 | sylancr 589 | . . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → ({〈1, 𝑀〉}‘1) = 𝑀) |
32 | 29, 31 | sylan9eqr 2878 | . . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝑀〉}‘𝑛) = 𝑀) |
33 | 32 | csbeq1d 3886 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
34 | 28 | fveq2d 6668 | . . . . . 6 ⊢ (𝑛 ∈ (1...1) → ({〈1, 𝐵〉}‘𝑛) = ({〈1, 𝐵〉}‘1)) |
35 | simpr 487 | . . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
36 | fvsng 6936 | . . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 𝐵 ∈ ℂ) → ({〈1, 𝐵〉}‘1) = 𝐵) | |
37 | 6, 35, 36 | sylancr 589 | . . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → ({〈1, 𝐵〉}‘1) = 𝐵) |
38 | 34, 37 | sylan9eqr 2878 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝐵〉}‘𝑛) = 𝐵) |
39 | 27, 33, 38 | 3eqtr4rd 2867 | . . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝐵〉}‘𝑛) = ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴) |
40 | 5, 7, 15, 26, 39 | fsum 15071 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 = (seq1( + , {〈1, 𝐵〉})‘1)) |
41 | 4, 40 | syl5eq 2868 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = (seq1( + , {〈1, 𝐵〉})‘1)) |
42 | 11, 37 | seq1i 13377 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → (seq1( + , {〈1, 𝐵〉})‘1) = 𝐵) |
43 | 41, 42 | eqtrd 2856 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Ⅎwnfc 2961 ⦋csb 3882 {csn 4560 〈cop 4566 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 1c1 10532 + caddc 10534 ℕcn 11632 ℤcz 11975 ...cfz 12886 seqcseq 13363 Σcsu 15036 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 |
This theorem is referenced by: fsumsplitsn 15094 sumsn 15095 |
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