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| Mirrors > Home > MPE Home > Th. List > prodeq1iOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of prodeq1i 15952 as of 1-Sep-2025. (Contributed by Scott Fenton, 4-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| prodeq1iOLD.1 | ⊢ 𝐴 = 𝐵 |
| Ref | Expression |
|---|---|
| prodeq1iOLD | ⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prodeq1iOLD.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | prodeq1 15943 | . 2 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∏cprod 15939 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seq 14043 df-prod 15940 |
| This theorem is referenced by: (None) |
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