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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem11 | Structured version Visualization version GIF version |
Description: A change of bound variable, often used in proofs for etransc 42445. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem11 | ⊢ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7153 | . . . . 5 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚)) | |
2 | 1 | oveq1d 7160 | . . . 4 ⊢ (𝑛 = 𝑚 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑚) ↑m (0...𝑀))) |
3 | 2 | rabeqdv 3482 | . . 3 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) |
4 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘)) | |
5 | 4 | cbvsumv 15041 | . . . . . . 7 ⊢ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) |
6 | fveq1 6662 | . . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑘) = (𝑑‘𝑘)) | |
7 | 6 | sumeq2sdv 15049 | . . . . . . 7 ⊢ (𝑐 = 𝑑 → Σ𝑘 ∈ (0...𝑀)(𝑐‘𝑘) = Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘)) |
8 | 5, 7 | syl5eq 2865 | . . . . . 6 ⊢ (𝑐 = 𝑑 → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘)) |
9 | 8 | eqeq1d 2820 | . . . . 5 ⊢ (𝑐 = 𝑑 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛)) |
10 | 9 | cbvrabv 3489 | . . . 4 ⊢ {𝑐 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛} |
11 | eqeq2 2830 | . . . . 5 ⊢ (𝑛 = 𝑚 → (Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛 ↔ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚)) | |
12 | 11 | rabbidv 3478 | . . . 4 ⊢ (𝑛 = 𝑚 → {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) |
13 | 10, 12 | syl5eq 2865 | . . 3 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) |
14 | 3, 13 | eqtrd 2853 | . 2 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) |
15 | 14 | cbvmptv 5160 | 1 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑑 ∈ ((0...𝑚) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 {crab 3139 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 0cc0 10525 ℕ0cn0 11885 ...cfz 12880 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-seq 13358 df-sum 15031 |
This theorem is referenced by: etransclem32 42428 etransclem33 42429 etransclem36 42432 etransclem37 42433 etransclem38 42434 etransclem40 42436 etransclem41 42437 etransclem42 42438 etransclem44 42440 etransclem45 42441 |
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