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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 15730 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 2898 | . . . 4 ⊢ Ⅎ𝑚𝐴 | |
2 | nfcsb1v 3917 | . . . 4 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐴 | |
3 | csbeq1a 3906 | . . . 4 ⊢ (𝑘 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑘⦌𝐴) | |
4 | 1, 2, 3 | cbvsumi 15681 | . . 3 ⊢ Σ𝑘 ∈ {𝑀}𝐴 = Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 |
5 | csbeq1 3895 | . . . 4 ⊢ (𝑚 = ({⟨1, 𝑀⟩}‘𝑛) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋({⟨1, 𝑀⟩}‘𝑛) / 𝑘⦌𝐴) | |
6 | 1nn 12259 | . . . . 5 ⊢ 1 ∈ ℕ | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → 1 ∈ ℕ) |
8 | simpl 481 | . . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → 𝑀 ∈ 𝑉) | |
9 | f1osng 6883 | . . . . . 6 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → {⟨1, 𝑀⟩}:{1}–1-1-onto→{𝑀}) | |
10 | 6, 8, 9 | sylancr 585 | . . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → {⟨1, 𝑀⟩}:{1}–1-1-onto→{𝑀}) |
11 | 1z 12628 | . . . . . 6 ⊢ 1 ∈ ℤ | |
12 | fzsn 13581 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
13 | f1oeq2 6831 | . . . . . 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 233 | . . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → {⟨1, 𝑀⟩}:(1...1)–1-1-onto→{𝑀}) |
16 | elsni 4647 | . . . . . . 7 ⊢ (𝑚 ∈ {𝑀} → 𝑚 = 𝑀) | |
17 | 16 | adantl 480 | . . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → 𝑚 = 𝑀) |
18 | 17 | csbeq1d 3896 | . . . . 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 3926 | . . . . . . 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 2828 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) |
26 | 18, 25 | eqeltrd 2828 | . . . 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 13550 | . . . . . . . 8 ⊢ (𝑛 ∈ (1...1) → 𝑛 = 1) | |
29 | 28 | fveq2d 6904 | . . . . . . 7 ⊢ (𝑛 ∈ (1...1) → ({⟨1, 𝑀⟩}‘𝑛) = ({⟨1, 𝑀⟩}‘1)) |
30 | fvsng 7193 | . . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → ({⟨1, 𝑀⟩}‘1) = 𝑀) | |
31 | 6, 8, 30 | sylancr 585 | . . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → ({⟨1, 𝑀⟩}‘1) = 𝑀) |
32 | 29, 31 | sylan9eqr 2789 | . . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ({⟨1, 𝑀⟩}‘𝑛) = 𝑀) |
33 | 32 | csbeq1d 3896 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ⦋({⟨1, 𝑀⟩}‘𝑛) / 𝑘⦌𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
34 | 28 | fveq2d 6904 | . . . . . 6 ⊢ (𝑛 ∈ (1...1) → ({⟨1, 𝐵⟩}‘𝑛) = ({⟨1, 𝐵⟩}‘1)) |
35 | simpr 483 | . . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
36 | fvsng 7193 | . . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 𝐵 ∈ ℂ) → ({⟨1, 𝐵⟩}‘1) = 𝐵) | |
37 | 6, 35, 36 | sylancr 585 | . . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → ({⟨1, 𝐵⟩}‘1) = 𝐵) |
38 | 34, 37 | sylan9eqr 2789 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ({⟨1, 𝐵⟩}‘𝑛) = 𝐵) |
39 | 27, 33, 38 | 3eqtr4rd 2778 | . . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ({⟨1, 𝐵⟩}‘𝑛) = ⦋({⟨1, 𝑀⟩}‘𝑛) / 𝑘⦌𝐴) |
40 | 5, 7, 15, 26, 39 | fsum 15704 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 = (seq1( + , {⟨1, 𝐵⟩})‘1)) |
41 | 4, 40 | eqtrid 2779 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = (seq1( + , {⟨1, 𝐵⟩})‘1)) |
42 | 11, 37 | seq1i 14018 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → (seq1( + , {⟨1, 𝐵⟩})‘1) = 𝐵) |
43 | 41, 42 | eqtrd 2767 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2878 ⦋csb 3892 {csn 4630 ⟨cop 4636 –1-1-onto→wf1o 6550 ‘cfv 6551 (class class class)co 7424 ℂcc 11142 1c1 11145 + caddc 11147 ℕcn 12248 ℤcz 12594 ...cfz 13522 seqcseq 14004 Σcsu 15670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-fz 13523 df-fzo 13666 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-sum 15671 |
This theorem is referenced by: fsumsplitsn 15728 sumsn 15730 |
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