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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 4717 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
4 | df-pr 4634 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})) |
6 | prfi 9361 | . . . 4 ⊢ {𝐴, 𝐵} ∈ Fin | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
8 | vex 3482 | . . . . 5 ⊢ 𝑘 ∈ V | |
9 | 8 | elpr 4655 | . . . 4 ⊢ (𝑘 ∈ {𝐴, 𝐵} ↔ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) |
10 | prodpr.1 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) | |
11 | 10 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 = 𝐸) |
12 | prodpr.e | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℂ) | |
13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐸 ∈ ℂ) |
14 | 11, 13 | eqeltrd 2839 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 ∈ ℂ) |
15 | prodpr.2 | . . . . . . 7 ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) | |
16 | 15 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 = 𝐹) |
17 | prodpr.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ℂ) | |
18 | 17 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐹 ∈ ℂ) |
19 | 16, 18 | eqeltrd 2839 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 ∈ ℂ) |
20 | 14, 19 | jaodan 959 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) → 𝐷 ∈ ℂ) |
21 | 9, 20 | sylan2b 594 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐷 ∈ ℂ) |
22 | 3, 5, 7, 21 | fprodsplit 15999 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (∏𝑘 ∈ {𝐴}𝐷 · ∏𝑘 ∈ {𝐵}𝐷)) |
23 | prodpr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
24 | 10 | prodsn 15995 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐸 ∈ ℂ) → ∏𝑘 ∈ {𝐴}𝐷 = 𝐸) |
25 | 23, 12, 24 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴}𝐷 = 𝐸) |
26 | prodpr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
27 | 15 | prodsn 15995 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐹 ∈ ℂ) → ∏𝑘 ∈ {𝐵}𝐷 = 𝐹) |
28 | 26, 17, 27 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐵}𝐷 = 𝐹) |
29 | 25, 28 | oveq12d 7449 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐴}𝐷 · ∏𝑘 ∈ {𝐵}𝐷) = (𝐸 · 𝐹)) |
30 | 22, 29 | eqtrd 2775 | 1 ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 {csn 4631 {cpr 4633 (class class class)co 7431 Fincfn 8984 ℂcc 11151 · cmul 11158 ∏cprod 15936 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-prod 15937 |
This theorem is referenced by: prodtp 32834 |
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