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Mirrors > Home > MPE Home > Th. List > fprodxp | Structured version Visualization version GIF version |
Description: Combine two products into a single product over the cartesian product. (Contributed by Scott Fenton, 1-Feb-2018.) |
Ref | Expression |
---|---|
fprodxp.1 | ⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝐷 = 𝐶) |
fprodxp.2 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodxp.3 | ⊢ (𝜑 → 𝐵 ∈ Fin) |
fprodxp.4 | ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
fprodxp | ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ (𝐴 × 𝐵)𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodxp.1 | . . 3 ⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝐷 = 𝐶) | |
2 | fprodxp.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
3 | fprodxp.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) |
5 | fprodxp.4 | . . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ) | |
6 | 1, 2, 4, 5 | fprod2d 15909 | . 2 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷) |
7 | iunxpconst 5741 | . . 3 ⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = (𝐴 × 𝐵) | |
8 | 7 | prodeq1i 15846 | . 2 ⊢ ∏𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐷 = ∏𝑧 ∈ (𝐴 × 𝐵)𝐷 |
9 | 6, 8 | eqtrdi 2788 | 1 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ (𝐴 × 𝐵)𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4623 〈cop 4629 ∪ ciun 4991 × cxp 5668 Fincfn 8924 ℂcc 11092 ∏cprod 15833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-inf2 9620 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 ax-pre-sup 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-sup 9421 df-oi 9489 df-card 9918 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-div 11856 df-nn 12197 df-2 12259 df-3 12260 df-n0 12457 df-z 12543 df-uz 12807 df-rp 12959 df-fz 13469 df-fzo 13612 df-seq 13951 df-exp 14012 df-hash 14275 df-cj 15030 df-re 15031 df-im 15032 df-sqrt 15166 df-abs 15167 df-clim 15416 df-prod 15834 |
This theorem is referenced by: (None) |
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