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Mirrors > Home > ILE Home > Th. List > prodeq1 | 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 2336 | . 2 ⊢ Ⅎ𝑘𝐴 | |
2 | nfcv 2336 | . 2 ⊢ Ⅎ𝑘𝐵 | |
3 | 1, 2 | prodeq1f 11698 | 1 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∏cprod 11696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3158 df-in 3160 df-ss 3167 df-if 3559 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-mpt 4093 df-cnv 4668 df-dm 4670 df-rn 4671 df-res 4672 df-iota 5216 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-recs 6360 df-frec 6446 df-seqfrec 10522 df-proddc 11697 |
This theorem is referenced by: prodeq1i 11707 prodeq1d 11710 prod1dc 11732 fprodf1o 11734 fprodssdc 11736 fprodmul 11737 fprodcl2lem 11751 fprodcllem 11752 fprodconst 11766 fprodap0 11767 fprod2d 11769 fprodrec 11775 fprodap0f 11782 fprodle 11786 fprodmodd 11787 |
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