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Mirrors > Home > MPE Home > Th. List > prodeq1i | Structured version Visualization version GIF version |
Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
prodeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
prodeq1i | ⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | prodeq1 15263 | . 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 1537 ∏cprod 15259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-iota 6314 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-seq 13371 df-prod 15260 |
This theorem is referenced by: prodeq12i 15274 fprodxp 15336 risefac0 15381 fallfacfwd 15390 prmo0 16372 breprexp 31904 etransclem31 42570 etransclem35 42574 hoidmv1le 42896 fmtnorec2 43725 |
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