Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prodtp | Structured version Visualization version GIF version |
Description: A product over a triple 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) |
prodtp.1 | ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) |
prodtp.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
prodtp.g | ⊢ (𝜑 → 𝐺 ∈ ℂ) |
prodtp.2 | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
prodtp.3 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Ref | Expression |
---|---|
prodtp | ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 · 𝐹) · 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodtp.2 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
2 | prodtp.3 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
3 | disjprsn 4643 | . . . 4 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) | |
4 | 1, 2, 3 | syl2anc 586 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
5 | df-tp 4565 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})) |
7 | tpfi 8787 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} ∈ Fin | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin) |
9 | vex 3494 | . . . . 5 ⊢ 𝑘 ∈ V | |
10 | 9 | eltp 4619 | . . . 4 ⊢ (𝑘 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵 ∨ 𝑘 = 𝐶)) |
11 | prodpr.1 | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) | |
12 | 11 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 = 𝐸) |
13 | prodpr.e | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℂ) | |
14 | 13 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐸 ∈ ℂ) |
15 | 12, 14 | eqeltrd 2912 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 ∈ ℂ) |
16 | 15 | adantlr 713 | . . . . 5 ⊢ (((𝜑 ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵 ∨ 𝑘 = 𝐶)) ∧ 𝑘 = 𝐴) → 𝐷 ∈ ℂ) |
17 | prodpr.2 | . . . . . . . 8 ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) | |
18 | 17 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 = 𝐹) |
19 | prodpr.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ℂ) | |
20 | 19 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐹 ∈ ℂ) |
21 | 18, 20 | eqeltrd 2912 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 ∈ ℂ) |
22 | 21 | adantlr 713 | . . . . 5 ⊢ (((𝜑 ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵 ∨ 𝑘 = 𝐶)) ∧ 𝑘 = 𝐵) → 𝐷 ∈ ℂ) |
23 | prodtp.1 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) | |
24 | 23 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐷 = 𝐺) |
25 | prodtp.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℂ) | |
26 | 25 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐺 ∈ ℂ) |
27 | 24, 26 | eqeltrd 2912 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐷 ∈ ℂ) |
28 | 27 | adantlr 713 | . . . . 5 ⊢ (((𝜑 ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵 ∨ 𝑘 = 𝐶)) ∧ 𝑘 = 𝐶) → 𝐷 ∈ ℂ) |
29 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵 ∨ 𝑘 = 𝐶)) → (𝑘 = 𝐴 ∨ 𝑘 = 𝐵 ∨ 𝑘 = 𝐶)) | |
30 | 16, 22, 28, 29 | mpjao3dan 1426 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵 ∨ 𝑘 = 𝐶)) → 𝐷 ∈ ℂ) |
31 | 10, 30 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵, 𝐶}) → 𝐷 ∈ ℂ) |
32 | 4, 6, 8, 31 | fprodsplit 15315 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = (∏𝑘 ∈ {𝐴, 𝐵}𝐷 · ∏𝑘 ∈ {𝐶}𝐷)) |
33 | prodpr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
34 | prodpr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
35 | prodpr.3 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
36 | 11, 17, 33, 34, 13, 19, 35 | prodpr 30542 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹)) |
37 | prodtp.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
38 | 23 | prodsn 15311 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐺 ∈ ℂ) → ∏𝑘 ∈ {𝐶}𝐷 = 𝐺) |
39 | 37, 25, 38 | syl2anc 586 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐷 = 𝐺) |
40 | 36, 39 | oveq12d 7167 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐴, 𝐵}𝐷 · ∏𝑘 ∈ {𝐶}𝐷) = ((𝐸 · 𝐹) · 𝐺)) |
41 | 32, 40 | eqtrd 2855 | 1 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 · 𝐹) · 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ w3o 1081 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∪ cun 3927 ∩ cin 3928 ∅c0 4284 {csn 4560 {cpr 4562 {ctp 4564 (class class class)co 7149 Fincfn 8502 ℂcc 10528 · cmul 10535 ∏cprod 15254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 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-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-clim 14840 df-prod 15255 |
This theorem is referenced by: hgt750lemg 31946 |
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