Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 14475. (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 2764 | . . . 4 ⊢ Ⅎ𝑚𝐴 | |
2 | nfcsb1v 3549 | . . . 4 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐴 | |
3 | csbeq1a 3542 | . . . 4 ⊢ (𝑘 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑘⦌𝐴) | |
4 | 1, 2, 3 | cbvsumi 14427 | . . 3 ⊢ Σ𝑘 ∈ {𝑀}𝐴 = Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 |
5 | csbeq1 3536 | . . . 4 ⊢ (𝑚 = ({〈1, 𝑀〉}‘𝑛) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴) | |
6 | 1nn 11031 | . . . . 5 ⊢ 1 ∈ ℕ | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 ∈ ℕ) |
8 | sumsnd.4 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
9 | f1osng 6177 | . . . . . 6 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀}) | |
10 | 6, 8, 9 | sylancr 695 | . . . . 5 ⊢ (𝜑 → {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀}) |
11 | 1z 11407 | . . . . . 6 ⊢ 1 ∈ ℤ | |
12 | fzsn 12383 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
13 | f1oeq2 6128 | . . . . . 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 224 | . . . 4 ⊢ (𝜑 → {〈1, 𝑀〉}:(1...1)–1-1-onto→{𝑀}) |
16 | elsni 4194 | . . . . . . 7 ⊢ (𝑚 ∈ {𝑀} → 𝑚 = 𝑀) | |
17 | 16 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → 𝑚 = 𝑀) |
18 | 17 | csbeq1d 3540 | . . . . 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 3554 | . . . . . . 7 ⊢ (𝜑 → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
23 | 22 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
24 | sumsnd.5 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
25 | 24 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → 𝐵 ∈ ℂ) |
26 | 23, 25 | eqeltrd 2701 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) |
27 | 18, 26 | eqeltrd 2701 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑚 / 𝑘⦌𝐴 ∈ ℂ) |
28 | 22 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
29 | elfz1eq 12352 | . . . . . . . 8 ⊢ (𝑛 ∈ (1...1) → 𝑛 = 1) | |
30 | 29 | fveq2d 6195 | . . . . . . 7 ⊢ (𝑛 ∈ (1...1) → ({〈1, 𝑀〉}‘𝑛) = ({〈1, 𝑀〉}‘1)) |
31 | fvsng 6447 | . . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → ({〈1, 𝑀〉}‘1) = 𝑀) | |
32 | 6, 8, 31 | sylancr 695 | . . . . . . 7 ⊢ (𝜑 → ({〈1, 𝑀〉}‘1) = 𝑀) |
33 | 30, 32 | sylan9eqr 2678 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝑀〉}‘𝑛) = 𝑀) |
34 | 33 | csbeq1d 3540 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
35 | 29 | fveq2d 6195 | . . . . . 6 ⊢ (𝑛 ∈ (1...1) → ({〈1, 𝐵〉}‘𝑛) = ({〈1, 𝐵〉}‘1)) |
36 | fvsng 6447 | . . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 𝐵 ∈ ℂ) → ({〈1, 𝐵〉}‘1) = 𝐵) | |
37 | 6, 24, 36 | sylancr 695 | . . . . . 6 ⊢ (𝜑 → ({〈1, 𝐵〉}‘1) = 𝐵) |
38 | 35, 37 | sylan9eqr 2678 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝐵〉}‘𝑛) = 𝐵) |
39 | 28, 34, 38 | 3eqtr4rd 2667 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝐵〉}‘𝑛) = ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴) |
40 | 5, 7, 15, 27, 39 | fsum 14451 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 = (seq1( + , {〈1, 𝐵〉})‘1)) |
41 | 4, 40 | syl5eq 2668 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = (seq1( + , {〈1, 𝐵〉})‘1)) |
42 | 11, 37 | seq1i 12815 | . 2 ⊢ (𝜑 → (seq1( + , {〈1, 𝐵〉})‘1) = 𝐵) |
43 | 41, 42 | eqtrd 2656 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ⦋csb 3533 {csn 4177 〈cop 4183 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 ℕcn 11020 ℤcz 11377 ...cfz 12326 seqcseq 12801 Σcsu 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 |
This theorem is referenced by: sumpair 39194 dvnmul 40158 sge0sn 40596 hoidmvlelem3 40811 |
Copyright terms: Public domain | W3C validator |