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Mirrors > Home > MPE Home > Th. List > prodeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.) |
Ref | Expression |
---|---|
prodeq1 | ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2751 | . 2 ⊢ Ⅎ𝑘𝐴 | |
2 | nfcv 2751 | . 2 ⊢ Ⅎ𝑘𝐵 | |
3 | 1, 2 | prodeq1f 14477 | 1 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∏cprod 14474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 df-prod 14475 |
This theorem is referenced by: prodeq1i 14487 prodeq1d 14490 prod1 14513 fprodf1o 14515 fprodss 14517 fprodcllem 14520 fprodmul 14529 fproddiv 14530 fprodconst 14547 fprodn0 14548 fprod2d 14550 fprodmodd 14567 coprmprod 15213 coprmproddvds 15215 fprodexp 38661 fprodabs2 38662 mccl 38665 fprodcn 38667 fprodcncf 38787 dvmptfprod 38835 dvnprodlem3 38838 hoidmvval 39467 ovnhoi 39493 hspmbllem2 39517 fmtnorec2 39993 |
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