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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 15944 | . 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 1539 ∏cprod 15940 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-xp 5690 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-iota 6513 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-seq 14044 df-prod 15941 | 
| This theorem is referenced by: prodeq12dv 15963 prodeq12rdv 15964 fprodf1o 15983 prodss 15984 fprod1 16000 fprodp1 16006 fprodfac 16010 fprodabs 16011 fprod2d 16018 fprodcom2 16021 risefacval 16045 fallfacval 16046 risefacval2 16047 fallfacval2 16048 risefacp1 16066 fallfacp1 16067 fallfacval4 16080 fprodefsum 16132 prmoval 17072 prmop1 17077 prmgapprmo 17101 gausslemma2dlem4 27414 breprexplema 34646 breprexplemc 34648 breprexp 34649 circlemethhgt 34659 bcprod 35739 aks4d1p1 42078 dvmptfprodlem 45964 dvmptfprod 45965 ovnval 46561 hoiprodp1 46608 hoidmv1le 46614 hspmbllem1 46646 fmtnorec2 47535 | 
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