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Mirrors > Home > MPE Home > Th. List > prodeq2dv | Structured version Visualization version GIF version |
Description: Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
prodeq2dv.1 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
prodeq2dv | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodeq2dv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) | |
2 | 1 | ralrimiva 3096 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) |
3 | 2 | prodeq2d 15368 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∏cprod 15351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-seq 13461 df-prod 15352 |
This theorem is referenced by: prodeq2sdv 15370 2cprodeq2dv 15371 prodeq12dv 15372 prodeq12rdv 15373 fprodf1o 15392 fprodss 15394 fprodsplit 15412 fprod2dlem 15426 risefallfac 15470 risefacfac 15481 fallfacfwd 15482 fproddvdsd 15780 prmgapprmo 16498 breprexplema 32180 breprexp 32183 breprexpnat 32184 vtsprod 32189 circlemethnat 32191 circlevma 32192 circlemethhgt 32193 hgt750lemg 32204 bcprod 33275 iprodgam 33279 aks4d1p1p1 39690 aks4d1p1p2 39697 aks4d1p1 39703 mccllem 42680 fprodcncf 42983 etransclem4 43321 etransclem13 43330 etransclem23 43340 etransclem31 43348 etransclem35 43352 hoicvrrex 43636 hsphoidmvle2 43665 hsphoidmvle 43666 hoidmvlelem2 43676 hoidmvlelem3 43677 hoidmvlelem4 43678 ovnhoilem1 43681 ovnhoilem2 43682 ovnhoi 43683 ovnlecvr2 43690 ovncvr2 43691 hspmbllem1 43706 hspmbl 43709 ovnovollem1 43736 vonioolem1 43760 vonicclem1 43763 vonn0icc 43768 vonn0ioo2 43770 vonsn 43771 vonn0icc2 43772 |
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