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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 2955 | . 2 ⊢ Ⅎ𝑘𝐴 | |
2 | nfcv 2955 | . 2 ⊢ Ⅎ𝑘𝐵 | |
3 | 1, 2 | prodeq1f 15254 | 1 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∏cprod 15251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-iota 6283 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seq 13365 df-prod 15252 |
This theorem is referenced by: prodeq1i 15264 prodeq1d 15267 prod1 15290 fprodf1o 15292 fprodss 15294 fprodcllem 15297 fprodmul 15306 fproddiv 15307 fprodconst 15324 fprodn0 15325 fprod2d 15327 fprodmodd 15343 coprmprod 15995 coprmproddvds 15997 fprodexp 42236 fprodabs2 42237 mccl 42240 fprodcn 42242 fprodcncf 42542 dvmptfprod 42587 dvnprodlem3 42590 hoidmvval 43216 ovnhoi 43242 hspmbllem2 43266 fmtnorec2 44060 |
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