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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 15103 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 2979 | . . . 4 ⊢ Ⅎ𝑚𝐴 | |
2 | nfcsb1v 3909 | . . . 4 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐴 | |
3 | csbeq1a 3899 | . . . 4 ⊢ (𝑘 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑘⦌𝐴) | |
4 | 1, 2, 3 | cbvsumi 15056 | . . 3 ⊢ Σ𝑘 ∈ {𝑀}𝐴 = Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 |
5 | csbeq1 3888 | . . . 4 ⊢ (𝑚 = ({〈1, 𝑀〉}‘𝑛) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴) | |
6 | 1nn 11651 | . . . . 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 6657 | . . . . . 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 12015 | . . . . . 6 ⊢ 1 ∈ ℤ | |
12 | fzsn 12952 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
13 | f1oeq2 6607 | . . . . . 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 4586 | . . . . . . 7 ⊢ (𝑚 ∈ {𝑀} → 𝑚 = 𝑀) | |
17 | 16 | adantl 484 | . . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → 𝑚 = 𝑀) |
18 | 17 | csbeq1d 3889 | . . . . 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 3918 | . . . . . . 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 2915 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) |
26 | 18, 25 | eqeltrd 2915 | . . . 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 12921 | . . . . . . . 8 ⊢ (𝑛 ∈ (1...1) → 𝑛 = 1) | |
29 | 28 | fveq2d 6676 | . . . . . . 7 ⊢ (𝑛 ∈ (1...1) → ({〈1, 𝑀〉}‘𝑛) = ({〈1, 𝑀〉}‘1)) |
30 | fvsng 6944 | . . . . . . . 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 2880 | . . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝑀〉}‘𝑛) = 𝑀) |
33 | 32 | csbeq1d 3889 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
34 | 28 | fveq2d 6676 | . . . . . 6 ⊢ (𝑛 ∈ (1...1) → ({〈1, 𝐵〉}‘𝑛) = ({〈1, 𝐵〉}‘1)) |
35 | simpr 487 | . . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
36 | fvsng 6944 | . . . . . . 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 2880 | . . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝐵〉}‘𝑛) = 𝐵) |
39 | 27, 33, 38 | 3eqtr4rd 2869 | . . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝐵〉}‘𝑛) = ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴) |
40 | 5, 7, 15, 26, 39 | fsum 15079 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 = (seq1( + , {〈1, 𝐵〉})‘1)) |
41 | 4, 40 | syl5eq 2870 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = (seq1( + , {〈1, 𝐵〉})‘1)) |
42 | 11, 37 | seq1i 13386 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → (seq1( + , {〈1, 𝐵〉})‘1) = 𝐵) |
43 | 41, 42 | eqtrd 2858 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 ⦋csb 3885 {csn 4569 〈cop 4575 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 1c1 10540 + caddc 10542 ℕcn 11640 ℤcz 11984 ...cfz 12895 seqcseq 13372 Σcsu 15044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 |
This theorem is referenced by: fsumsplitsn 15102 sumsn 15103 |
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