MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fprodss Structured version   Visualization version   GIF version

Theorem fprodss 15836
Description: Change the index set to a subset in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)
Hypotheses
Ref Expression
fprodss.1 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
fprodss.2 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
fprodss.3 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 1)
fprodss.4 (πœ‘ β†’ 𝐡 ∈ Fin)
Assertion
Ref Expression
fprodss (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
Distinct variable groups:   𝐴,π‘˜   𝐡,π‘˜   πœ‘,π‘˜
Allowed substitution hint:   𝐢(π‘˜)

Proof of Theorem fprodss
Dummy variables 𝑓 π‘š 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fprodss.1 . . 3 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
2 sseq2 3971 . . . . 5 (𝐡 = βˆ… β†’ (𝐴 βŠ† 𝐡 ↔ 𝐴 βŠ† βˆ…))
3 ss0 4359 . . . . 5 (𝐴 βŠ† βˆ… β†’ 𝐴 = βˆ…)
42, 3syl6bi 253 . . . 4 (𝐡 = βˆ… β†’ (𝐴 βŠ† 𝐡 β†’ 𝐴 = βˆ…))
5 prodeq1 15797 . . . . . 6 (𝐴 = βˆ… β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ βˆ… 𝐢)
6 prodeq1 15797 . . . . . . 7 (𝐡 = βˆ… β†’ βˆπ‘˜ ∈ 𝐡 𝐢 = βˆπ‘˜ ∈ βˆ… 𝐢)
76eqcomd 2739 . . . . . 6 (𝐡 = βˆ… β†’ βˆπ‘˜ ∈ βˆ… 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
85, 7sylan9eq 2793 . . . . 5 ((𝐴 = βˆ… ∧ 𝐡 = βˆ…) β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
98expcom 415 . . . 4 (𝐡 = βˆ… β†’ (𝐴 = βˆ… β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢))
104, 9syld 47 . . 3 (𝐡 = βˆ… β†’ (𝐴 βŠ† 𝐡 β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢))
111, 10syl5com 31 . 2 (πœ‘ β†’ (𝐡 = βˆ… β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢))
12 cnvimass 6034 . . . . . . . . 9 (◑𝑓 β€œ 𝐴) βŠ† dom 𝑓
13 simprr 772 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)
14 f1of 6785 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))⟢𝐡)
1513, 14syl 17 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))⟢𝐡)
1612, 15fssdm 6689 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (◑𝑓 β€œ 𝐴) βŠ† (1...(β™―β€˜π΅)))
17 f1ofn 6786 . . . . . . . . . . . 12 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓 Fn (1...(β™―β€˜π΅)))
18 elpreima 7009 . . . . . . . . . . . 12 (𝑓 Fn (1...(β™―β€˜π΅)) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴)))
1913, 17, 183syl 18 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴)))
2015ffvelcdmda 7036 . . . . . . . . . . . . 13 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (1...(β™―β€˜π΅))) β†’ (π‘“β€˜π‘›) ∈ 𝐡)
2120ex 414 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑛 ∈ (1...(β™―β€˜π΅)) β†’ (π‘“β€˜π‘›) ∈ 𝐡))
2221adantrd 493 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ ((𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴) β†’ (π‘“β€˜π‘›) ∈ 𝐡))
2319, 22sylbid 239 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) β†’ (π‘“β€˜π‘›) ∈ 𝐡))
2423imp 408 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (◑𝑓 β€œ 𝐴)) β†’ (π‘“β€˜π‘›) ∈ 𝐡)
25 fprodss.2 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
2625ex 414 . . . . . . . . . . . . . 14 (πœ‘ β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
2726adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
28 eldif 3921 . . . . . . . . . . . . . . 15 (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↔ (π‘˜ ∈ 𝐡 ∧ Β¬ π‘˜ ∈ 𝐴))
29 fprodss.3 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 1)
30 ax-1cn 11114 . . . . . . . . . . . . . . . 16 1 ∈ β„‚
3129, 30eqeltrdi 2842 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ β„‚)
3228, 31sylan2br 596 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘˜ ∈ 𝐡 ∧ Β¬ π‘˜ ∈ 𝐴)) β†’ 𝐢 ∈ β„‚)
3332expr 458 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (Β¬ π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
3427, 33pm2.61d 179 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)
3534adantlr 714 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘˜ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)
3635fmpttd 7064 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (π‘˜ ∈ 𝐡 ↦ 𝐢):π΅βŸΆβ„‚)
3736ffvelcdmda 7036 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ (π‘“β€˜π‘›) ∈ 𝐡) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ β„‚)
3824, 37syldan 592 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (◑𝑓 β€œ 𝐴)) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ β„‚)
39 eqid 2733 . . . . . . . . 9 (β„€β‰₯β€˜1) = (β„€β‰₯β€˜1)
40 simprl 770 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (β™―β€˜π΅) ∈ β„•)
41 nnuz 12811 . . . . . . . . . 10 β„• = (β„€β‰₯β€˜1)
4240, 41eleqtrdi 2844 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (β™―β€˜π΅) ∈ (β„€β‰₯β€˜1))
43 ssidd 3968 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (1...(β™―β€˜π΅)) βŠ† (1...(β™―β€˜π΅)))
4439, 42, 43fprodntriv 15830 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆƒπ‘š ∈ (β„€β‰₯β€˜1)βˆƒπ‘¦(𝑦 β‰  0 ∧ seqπ‘š( Β· , (𝑛 ∈ (β„€β‰₯β€˜1) ↦ if(𝑛 ∈ (1...(β™―β€˜π΅)), ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)), 1))) ⇝ 𝑦))
45 eldifi 4087 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ 𝑛 ∈ (1...(β™―β€˜π΅)))
4645, 20sylan2 594 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (π‘“β€˜π‘›) ∈ 𝐡)
47 eldifn 4088 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ Β¬ 𝑛 ∈ (◑𝑓 β€œ 𝐴))
4847adantl 483 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ Β¬ 𝑛 ∈ (◑𝑓 β€œ 𝐴))
4945adantl 483 . . . . . . . . . . . . 13 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ 𝑛 ∈ (1...(β™―β€˜π΅)))
5019adantr 482 . . . . . . . . . . . . 13 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴)))
5149, 50mpbirand 706 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) ↔ (π‘“β€˜π‘›) ∈ 𝐴))
5248, 51mtbid 324 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ Β¬ (π‘“β€˜π‘›) ∈ 𝐴)
5346, 52eldifd 3922 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (π‘“β€˜π‘›) ∈ (𝐡 βˆ– 𝐴))
54 difss 4092 . . . . . . . . . . . . 13 (𝐡 βˆ– 𝐴) βŠ† 𝐡
55 resmpt 5992 . . . . . . . . . . . . 13 ((𝐡 βˆ– 𝐴) βŠ† 𝐡 β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴)) = (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢))
5654, 55ax-mp 5 . . . . . . . . . . . 12 ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴)) = (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)
5756fveq1i 6844 . . . . . . . . . . 11 (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴))β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›))
58 fvres 6862 . . . . . . . . . . 11 ((π‘“β€˜π‘›) ∈ (𝐡 βˆ– 𝐴) β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴))β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
5957, 58eqtr3id 2787 . . . . . . . . . 10 ((π‘“β€˜π‘›) ∈ (𝐡 βˆ– 𝐴) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
6053, 59syl 17 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
61 1ex 11156 . . . . . . . . . . . . . . 15 1 ∈ V
6261elsn2 4626 . . . . . . . . . . . . . 14 (𝐢 ∈ {1} ↔ 𝐢 = 1)
6329, 62sylibr 233 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ {1})
6463fmpttd 7064 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢):(𝐡 βˆ– 𝐴)⟢{1})
6564ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢):(𝐡 βˆ– 𝐴)⟢{1})
6665, 53ffvelcdmd 7037 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ {1})
67 elsni 4604 . . . . . . . . . 10 (((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ {1} β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = 1)
6866, 67syl 17 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = 1)
6960, 68eqtr3d 2775 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = 1)
70 fzssuz 13488 . . . . . . . . 9 (1...(β™―β€˜π΅)) βŠ† (β„€β‰₯β€˜1)
7170a1i 11 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (1...(β™―β€˜π΅)) βŠ† (β„€β‰₯β€˜1))
7216, 38, 44, 69, 71prodss 15835 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆπ‘› ∈ (◑𝑓 β€œ 𝐴)((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = βˆπ‘› ∈ (1...(β™―β€˜π΅))((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
731adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝐴 βŠ† 𝐡)
7473resmptd 5995 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴) = (π‘˜ ∈ 𝐴 ↦ 𝐢))
7574fveq1d 6845 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴)β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š))
76 fvres 6862 . . . . . . . . . 10 (π‘š ∈ 𝐴 β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
7775, 76sylan9req 2794 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
7877prodeq2dv 15811 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
79 fveq2 6843 . . . . . . . . 9 (π‘š = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
80 fzfid 13884 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (1...(β™―β€˜π΅)) ∈ Fin)
8180, 15fisuppfi 9317 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (◑𝑓 β€œ 𝐴) ∈ Fin)
82 f1of1 6784 . . . . . . . . . . . 12 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))–1-1→𝐡)
8313, 82syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))–1-1→𝐡)
84 f1ores 6799 . . . . . . . . . . 11 ((𝑓:(1...(β™―β€˜π΅))–1-1→𝐡 ∧ (◑𝑓 β€œ 𝐴) βŠ† (1...(β™―β€˜π΅))) β†’ (𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-ontoβ†’(𝑓 β€œ (◑𝑓 β€œ 𝐴)))
8583, 16, 84syl2anc 585 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-ontoβ†’(𝑓 β€œ (◑𝑓 β€œ 𝐴)))
86 f1ofo 6792 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))–onto→𝐡)
8713, 86syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))–onto→𝐡)
88 foimacnv 6802 . . . . . . . . . . . 12 ((𝑓:(1...(β™―β€˜π΅))–onto→𝐡 ∧ 𝐴 βŠ† 𝐡) β†’ (𝑓 β€œ (◑𝑓 β€œ 𝐴)) = 𝐴)
8987, 73, 88syl2anc 585 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑓 β€œ (◑𝑓 β€œ 𝐴)) = 𝐴)
9089f1oeq3d 6782 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ ((𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-ontoβ†’(𝑓 β€œ (◑𝑓 β€œ 𝐴)) ↔ (𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-onto→𝐴))
9185, 90mpbid 231 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-onto→𝐴)
92 fvres 6862 . . . . . . . . . 10 (𝑛 ∈ (◑𝑓 β€œ 𝐴) β†’ ((𝑓 β†Ύ (◑𝑓 β€œ 𝐴))β€˜π‘›) = (π‘“β€˜π‘›))
9392adantl 483 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (◑𝑓 β€œ 𝐴)) β†’ ((𝑓 β†Ύ (◑𝑓 β€œ 𝐴))β€˜π‘›) = (π‘“β€˜π‘›))
9473sselda 3945 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ π‘š ∈ 𝐡)
9536ffvelcdmda 7036 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐡) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) ∈ β„‚)
9694, 95syldan 592 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) ∈ β„‚)
9779, 81, 91, 93, 96fprodf1o 15834 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = βˆπ‘› ∈ (◑𝑓 β€œ 𝐴)((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
9878, 97eqtrd 2773 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = βˆπ‘› ∈ (◑𝑓 β€œ 𝐴)((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
99 eqidd 2734 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (1...(β™―β€˜π΅))) β†’ (π‘“β€˜π‘›) = (π‘“β€˜π‘›))
10079, 80, 13, 99, 95fprodf1o 15834 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆπ‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = βˆπ‘› ∈ (1...(β™―β€˜π΅))((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
10172, 98, 1003eqtr4d 2783 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = βˆπ‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
102 prodfc 15833 . . . . . 6 βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = βˆπ‘˜ ∈ 𝐴 𝐢
103 prodfc 15833 . . . . . 6 βˆπ‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = βˆπ‘˜ ∈ 𝐡 𝐢
104101, 102, 1033eqtr3g 2796 . . . . 5 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
105104expr 458 . . . 4 ((πœ‘ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢))
106105exlimdv 1937 . . 3 ((πœ‘ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢))
107106expimpd 455 . 2 (πœ‘ β†’ (((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡) β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢))
108 fprodss.4 . . 3 (πœ‘ β†’ 𝐡 ∈ Fin)
109 fz1f1o 15600 . . 3 (𝐡 ∈ Fin β†’ (𝐡 = βˆ… ∨ ((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)))
110108, 109syl 17 . 2 (πœ‘ β†’ (𝐡 = βˆ… ∨ ((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)))
11111, 107, 110mpjaod 859 1 (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   βˆ– cdif 3908   βŠ† wss 3911  βˆ…c0 4283  {csn 4587   ↦ cmpt 5189  β—‘ccnv 5633   β†Ύ cres 5636   β€œ cima 5637   Fn wfn 6492  βŸΆwf 6493  β€“1-1β†’wf1 6494  β€“ontoβ†’wfo 6495  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  Fincfn 8886  β„‚cc 11054  1c1 11057  β„•cn 12158  β„€β‰₯cuz 12768  ...cfz 13430  β™―chash 14236  βˆcprod 15793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-prod 15794
This theorem is referenced by:  fprodsplit  15854
  Copyright terms: Public domain W3C validator