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Mirrors > Home > MPE Home > Th. List > prodrb | Structured version Visualization version GIF version |
Description: Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
prodmo.1 | ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) |
prodmo.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
prodrb.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
prodrb.5 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
prodrb.6 | ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) |
prodrb.7 | ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑁)) |
Ref | Expression |
---|---|
prodrb | ⊢ (𝜑 → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodmo.1 | . . 3 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) | |
2 | prodmo.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
3 | prodrb.4 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | prodrb.5 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
5 | prodrb.6 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) | |
6 | prodrb.7 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑁)) | |
7 | 1, 2, 3, 4, 5, 6 | prodrblem2 15284 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶)) |
8 | 1, 2, 4, 3, 6, 5 | prodrblem2 15284 | . . 3 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → (seq𝑁( · , 𝐹) ⇝ 𝐶 ↔ seq𝑀( · , 𝐹) ⇝ 𝐶)) |
9 | 8 | bicomd 225 | . 2 ⊢ ((𝜑 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶)) |
10 | uztric 12265 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ∨ 𝑀 ∈ (ℤ≥‘𝑁))) | |
11 | 3, 4, 10 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘𝑀) ∨ 𝑀 ∈ (ℤ≥‘𝑁))) |
12 | 7, 9, 11 | mpjaodan 955 | 1 ⊢ (𝜑 → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ifcif 4466 class class class wbr 5065 ↦ cmpt 5145 ‘cfv 6354 ℂcc 10534 1c1 10537 · cmul 10541 ℤcz 11980 ℤ≥cuz 12242 seqcseq 13368 ⇝ cli 14840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-seq 13369 df-clim 14844 |
This theorem is referenced by: prodmo 15289 zprod 15290 |
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