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Mirrors > Home > MPE Home > Th. List > prodeq1d | Structured version Visualization version GIF version |
Description: Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
prodeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
prodeq1d | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | prodeq1 15471 | . 2 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∏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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-xp 5557 df-cnv 5559 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-iota 6338 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-seq 13575 df-prod 15468 |
This theorem is referenced by: prodeq12dv 15488 prodeq12rdv 15489 fprodf1o 15508 prodss 15509 fprod1 15525 fprodp1 15531 fprodfac 15535 fprodabs 15536 fprod2d 15543 fprodcom2 15546 risefacval 15570 fallfacval 15571 risefacval2 15572 fallfacval2 15573 risefacp1 15591 fallfacp1 15592 fallfacval4 15605 fprodefsum 15656 prmoval 16586 prmop1 16591 prmgapprmo 16615 gausslemma2dlem4 26250 breprexplema 32322 breprexplemc 32324 breprexp 32325 circlemethhgt 32335 bcprod 33422 aks4d1p1 39817 dvmptfprodlem 43160 dvmptfprod 43161 ovnval 43754 hoiprodp1 43801 hoidmv1le 43807 hspmbllem1 43839 fmtnorec2 44668 |
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