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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 2974 | . 2 ⊢ Ⅎ𝑘𝐴 | |
2 | nfcv 2974 | . 2 ⊢ Ⅎ𝑘𝐵 | |
3 | 1, 2 | prodeq1f 15250 | 1 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∏cprod 15247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-iota 6307 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-seq 13358 df-prod 15248 |
This theorem is referenced by: prodeq1i 15260 prodeq1d 15263 prod1 15286 fprodf1o 15288 fprodss 15290 fprodcllem 15293 fprodmul 15302 fproddiv 15303 fprodconst 15320 fprodn0 15321 fprod2d 15323 fprodmodd 15339 coprmprod 15993 coprmproddvds 15995 fprodexp 41751 fprodabs2 41752 mccl 41755 fprodcn 41757 fprodcncf 42060 dvmptfprod 42106 dvnprodlem3 42109 hoidmvval 42736 ovnhoi 42762 hspmbllem2 42786 fmtnorec2 43582 |
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