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 4628 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
4 | df-pr 4544 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})) |
6 | prfi 8946 | . . . 4 ⊢ {𝐴, 𝐵} ∈ Fin | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
8 | vex 3412 | . . . . 5 ⊢ 𝑘 ∈ V | |
9 | 8 | elpr 4564 | . . . 4 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↔ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) |
10 | prodpr.1 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) | |
11 | 10 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 = 𝐸) |
12 | prodpr.e | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℂ) | |
13 | 12 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐸 ∈ ℂ) |
14 | 11, 13 | eqeltrd 2838 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 ∈ ℂ) |
15 | prodpr.2 | . . . . . . 7 ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) | |
16 | 15 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 = 𝐹) |
17 | prodpr.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ℂ) | |
18 | 17 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐹 ∈ ℂ) |
19 | 16, 18 | eqeltrd 2838 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 ∈ ℂ) |
20 | 14, 19 | jaodan 958 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) → 𝐷 ∈ ℂ) |
21 | 9, 20 | sylan2b 597 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐷 ∈ ℂ) |
22 | 3, 5, 7, 21 | fprodsplit 15528 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (∏𝑘 ∈ {𝐴}𝐷 · ∏𝑘 ∈ {𝐵}𝐷)) |
23 | prodpr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
24 | 10 | prodsn 15524 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐸 ∈ ℂ) → ∏𝑘 ∈ {𝐴}𝐷 = 𝐸) |
25 | 23, 12, 24 | syl2anc 587 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴}𝐷 = 𝐸) |
26 | prodpr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
27 | 15 | prodsn 15524 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐹 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐷 = 𝐹) |
28 | 26, 17, 27 | syl2anc 587 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐷 = 𝐹) |
29 | 25, 28 | oveq12d 7231 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐴}𝐷 · ∏𝑘 ∈ {𝐵}𝐷) = (𝐸 · 𝐹)) |
30 | 22, 29 | eqtrd 2777 | 1 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∪ cun 3864 ∩ cin 3865 ∅c0 4237 {csn 4541 {cpr 4543 (class class class)co 7213 Fincfn 8626 ℂcc 10727 · cmul 10734 ∏cprod 15467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-prod 15468 |
This theorem is referenced by: prodtp 30861 |
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