Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prodpr | Structured version Visualization version GIF version |
Description: A product over a pair is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
Ref | Expression |
---|---|
prodpr.1 | ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) |
prodpr.2 | ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) |
prodpr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
prodpr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
prodpr.e | ⊢ (𝜑 → 𝐸 ∈ ℂ) |
prodpr.f | ⊢ (𝜑 → 𝐹 ∈ ℂ) |
prodpr.3 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
prodpr | ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodpr.3 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | disjsn2 4650 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
4 | df-pr 4572 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})) |
6 | prfi 8795 | . . . 4 ⊢ {𝐴, 𝐵} ∈ Fin | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
8 | vex 3499 | . . . . 5 ⊢ 𝑘 ∈ V | |
9 | 8 | elpr 4592 | . . . 4 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↔ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) |
10 | prodpr.1 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) | |
11 | 10 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 = 𝐸) |
12 | prodpr.e | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℂ) | |
13 | 12 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐸 ∈ ℂ) |
14 | 11, 13 | eqeltrd 2915 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 ∈ ℂ) |
15 | prodpr.2 | . . . . . . 7 ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) | |
16 | 15 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 = 𝐹) |
17 | prodpr.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ℂ) | |
18 | 17 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐹 ∈ ℂ) |
19 | 16, 18 | eqeltrd 2915 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 ∈ ℂ) |
20 | 14, 19 | jaodan 954 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) → 𝐷 ∈ ℂ) |
21 | 9, 20 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐷 ∈ ℂ) |
22 | 3, 5, 7, 21 | fprodsplit 15322 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (∏𝑘 ∈ {𝐴}𝐷 · ∏𝑘 ∈ {𝐵}𝐷)) |
23 | prodpr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
24 | 10 | prodsn 15318 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐸 ∈ ℂ) → ∏𝑘 ∈ {𝐴}𝐷 = 𝐸) |
25 | 23, 12, 24 | syl2anc 586 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴}𝐷 = 𝐸) |
26 | prodpr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
27 | 15 | prodsn 15318 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐹 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐷 = 𝐹) |
28 | 26, 17, 27 | syl2anc 586 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐷 = 𝐹) |
29 | 25, 28 | oveq12d 7176 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐴}𝐷 · ∏𝑘 ∈ {𝐵}𝐷) = (𝐸 · 𝐹)) |
30 | 22, 29 | eqtrd 2858 | 1 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 {cpr 4571 (class class class)co 7158 Fincfn 8511 ℂcc 10537 · cmul 10544 ∏cprod 15261 |
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-fal 1550 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-prod 15262 |
This theorem is referenced by: prodtp 30545 |
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