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Mirrors > Home > MPE Home > Th. List > Mathboxes > sumsnd | Structured version Visualization version GIF version |
Description: A sum of a singleton is the term. The deduction version of sumsn 15694. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
sumsnd.1 | ⊢ (𝜑 → Ⅎ𝑘𝐵) |
sumsnd.2 | ⊢ Ⅎ𝑘𝜑 |
sumsnd.3 | ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐵) |
sumsnd.4 | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
sumsnd.5 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
sumsnd | ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2895 | . . . 4 ⊢ Ⅎ𝑚𝐴 | |
2 | nfcsb1v 3911 | . . . 4 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐴 | |
3 | csbeq1a 3900 | . . . 4 ⊢ (𝑘 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑘⦌𝐴) | |
4 | 1, 2, 3 | cbvsumi 15645 | . . 3 ⊢ Σ𝑘 ∈ {𝑀}𝐴 = Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 |
5 | csbeq1 3889 | . . . 4 ⊢ (𝑚 = ({⟨1, 𝑀⟩}‘𝑛) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋({⟨1, 𝑀⟩}‘𝑛) / 𝑘⦌𝐴) | |
6 | 1nn 12222 | . . . . 5 ⊢ 1 ∈ ℕ | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 ∈ ℕ) |
8 | sumsnd.4 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
9 | f1osng 6865 | . . . . . 6 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → {⟨1, 𝑀⟩}:{1}–1-1-onto→{𝑀}) | |
10 | 6, 8, 9 | sylancr 586 | . . . . 5 ⊢ (𝜑 → {⟨1, 𝑀⟩}:{1}–1-1-onto→{𝑀}) |
11 | 1z 12591 | . . . . . 6 ⊢ 1 ∈ ℤ | |
12 | fzsn 13544 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
13 | f1oeq2 6813 | . . . . . 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 ⊢ (𝜑 → {⟨1, 𝑀⟩}:(1...1)–1-1-onto→{𝑀}) |
16 | elsni 4638 | . . . . . . 7 ⊢ (𝑚 ∈ {𝑀} → 𝑚 = 𝑀) | |
17 | 16 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → 𝑚 = 𝑀) |
18 | 17 | csbeq1d 3890 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
19 | sumsnd.2 | . . . . . . . 8 ⊢ Ⅎ𝑘𝜑 | |
20 | sumsnd.1 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑘𝐵) | |
21 | sumsnd.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐵) | |
22 | 19, 20, 8, 21 | csbiedf 3917 | . . . . . . 7 ⊢ (𝜑 → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
23 | 22 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
24 | sumsnd.5 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
25 | 24 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → 𝐵 ∈ ℂ) |
26 | 23, 25 | eqeltrd 2825 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) |
27 | 18, 26 | eqeltrd 2825 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑚 / 𝑘⦌𝐴 ∈ ℂ) |
28 | 22 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
29 | elfz1eq 13513 | . . . . . . . 8 ⊢ (𝑛 ∈ (1...1) → 𝑛 = 1) | |
30 | 29 | fveq2d 6886 | . . . . . . 7 ⊢ (𝑛 ∈ (1...1) → ({⟨1, 𝑀⟩}‘𝑛) = ({⟨1, 𝑀⟩}‘1)) |
31 | fvsng 7171 | . . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → ({⟨1, 𝑀⟩}‘1) = 𝑀) | |
32 | 6, 8, 31 | sylancr 586 | . . . . . . 7 ⊢ (𝜑 → ({⟨1, 𝑀⟩}‘1) = 𝑀) |
33 | 30, 32 | sylan9eqr 2786 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ({⟨1, 𝑀⟩}‘𝑛) = 𝑀) |
34 | 33 | csbeq1d 3890 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ⦋({⟨1, 𝑀⟩}‘𝑛) / 𝑘⦌𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
35 | 29 | fveq2d 6886 | . . . . . 6 ⊢ (𝑛 ∈ (1...1) → ({⟨1, 𝐵⟩}‘𝑛) = ({⟨1, 𝐵⟩}‘1)) |
36 | fvsng 7171 | . . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 𝐵 ∈ ℂ) → ({⟨1, 𝐵⟩}‘1) = 𝐵) | |
37 | 6, 24, 36 | sylancr 586 | . . . . . 6 ⊢ (𝜑 → ({⟨1, 𝐵⟩}‘1) = 𝐵) |
38 | 35, 37 | sylan9eqr 2786 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ({⟨1, 𝐵⟩}‘𝑛) = 𝐵) |
39 | 28, 34, 38 | 3eqtr4rd 2775 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ({⟨1, 𝐵⟩}‘𝑛) = ⦋({⟨1, 𝑀⟩}‘𝑛) / 𝑘⦌𝐴) |
40 | 5, 7, 15, 27, 39 | fsum 15668 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 = (seq1( + , {⟨1, 𝐵⟩})‘1)) |
41 | 4, 40 | eqtrid 2776 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = (seq1( + , {⟨1, 𝐵⟩})‘1)) |
42 | 11, 37 | seq1i 13981 | . 2 ⊢ (𝜑 → (seq1( + , {⟨1, 𝐵⟩})‘1) = 𝐵) |
43 | 41, 42 | eqtrd 2764 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2875 ⦋csb 3886 {csn 4621 ⟨cop 4627 –1-1-onto→wf1o 6533 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 1c1 11108 + caddc 11110 ℕcn 12211 ℤcz 12557 ...cfz 13485 seqcseq 13967 Σcsu 15634 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fz 13486 df-fzo 13629 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 |
This theorem is referenced by: sumpair 44269 dvnmul 45205 sge0sn 45641 hoidmvlelem3 45859 |
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