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Mirrors > Home > ILE Home > Th. List > prodeq1d | 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 11480 | . 2 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∏cprod 11477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-recs 6264 df-frec 6350 df-seqfrec 10371 df-proddc 11478 |
This theorem is referenced by: prodeq12dv 11496 prodeq12rdv 11497 fprodf1o 11515 fprod1 11521 fprodp1 11527 fprodcl2lem 11532 fprodfac 11542 fprodabs 11543 fprod2d 11550 fprodcom2fi 11553 eulerthlemrprm 12140 eulerthlema 12141 |
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