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Theorem fprodssdc 11611
Description: Change the index set to a subset in a finite sum. (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„‚)
fprodssdc.a (πœ‘ β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)
fprodss.3 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 1)
fprodss.4 (πœ‘ β†’ 𝐡 ∈ Fin)
Assertion
Ref Expression
fprodssdc (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
Distinct variable groups:   𝐴,𝑗,π‘˜   𝐡,π‘˜,𝑗   πœ‘,π‘˜   𝑗,π‘˜
Allowed substitution hints:   πœ‘(𝑗)   𝐢(𝑗,π‘˜)

Proof of Theorem fprodssdc
Dummy variables 𝑓 𝑝 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 3191 . . . . 5 (𝐡 = βˆ… β†’ (𝐴 βŠ† 𝐡 ↔ 𝐴 βŠ† βˆ…))
3 ss0 3475 . . . . 5 (𝐴 βŠ† βˆ… β†’ 𝐴 = βˆ…)
42, 3biimtrdi 163 . . . 4 (𝐡 = βˆ… β†’ (𝐴 βŠ† 𝐡 β†’ 𝐴 = βˆ…))
5 prodeq1 11574 . . . . . 6 (𝐴 = βˆ… β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ βˆ… 𝐢)
6 prodeq1 11574 . . . . . . 7 (𝐡 = βˆ… β†’ βˆπ‘˜ ∈ 𝐡 𝐢 = βˆπ‘˜ ∈ βˆ… 𝐢)
76eqcomd 2193 . . . . . 6 (𝐡 = βˆ… β†’ βˆπ‘˜ ∈ βˆ… 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
85, 7sylan9eq 2240 . . . . 5 ((𝐴 = βˆ… ∧ 𝐡 = βˆ…) β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
98expcom 116 . . . 4 (𝐡 = βˆ… β†’ (𝐴 = βˆ… β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢))
104, 9syld 45 . . 3 (𝐡 = βˆ… β†’ (𝐴 βŠ† 𝐡 β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢))
111, 10syl5com 29 . 2 (πœ‘ β†’ (𝐡 = βˆ… β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢))
12 cnvimass 5003 . . . . . . . . 9 (◑𝑓 β€œ 𝐴) βŠ† dom 𝑓
13 simprr 531 . . . . . . . . . 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 5473 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))⟢𝐡)
1513, 14syl 14 . . . . . . . . 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 5392 . . . . . . . 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 5474 . . . . . . . . . . . 12 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓 Fn (1...(β™―β€˜π΅)))
18 elpreima 5648 . . . . . . . . . . . 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 17 . . . . . . . . . . 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 5664 . . . . . . . . . . . . 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 115 . . . . . . . . . . . 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 279 . . . . . . . . . . 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 150 . . . . . . . . . 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 124 . . . . . . . . 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 115 . . . . . . . . . . . . . 14 (πœ‘ β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
2726adantr 276 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
28 eldif 3150 . . . . . . . . . . . . . . 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 7917 . . . . . . . . . . . . . . . 16 1 ∈ β„‚
3129, 30eqeltrdi 2278 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ β„‚)
3228, 31sylan2br 288 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘˜ ∈ 𝐡 ∧ Β¬ π‘˜ ∈ 𝐴)) β†’ 𝐢 ∈ β„‚)
3332expr 375 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (Β¬ π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
34 eleq1 2250 . . . . . . . . . . . . . . . 16 (𝑗 = π‘˜ β†’ (𝑗 ∈ 𝐴 ↔ π‘˜ ∈ 𝐴))
3534dcbid 839 . . . . . . . . . . . . . . 15 (𝑗 = π‘˜ β†’ (DECID 𝑗 ∈ 𝐴 ↔ DECID π‘˜ ∈ 𝐴))
36 fprodssdc.a . . . . . . . . . . . . . . . 16 (πœ‘ β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)
3736adantr 276 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)
38 simpr 110 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ π‘˜ ∈ 𝐡)
3935, 37, 38rspcdva 2858 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ DECID π‘˜ ∈ 𝐴)
40 exmiddc 837 . . . . . . . . . . . . . 14 (DECID π‘˜ ∈ 𝐴 β†’ (π‘˜ ∈ 𝐴 ∨ Β¬ π‘˜ ∈ 𝐴))
4139, 40syl 14 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (π‘˜ ∈ 𝐴 ∨ Β¬ π‘˜ ∈ 𝐴))
4227, 33, 41mpjaod 719 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)
4342adantlr 477 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘˜ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)
4443fmpttd 5684 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (π‘˜ ∈ 𝐡 ↦ 𝐢):π΅βŸΆβ„‚)
4544ffvelcdmda 5664 . . . . . . . . 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„‚)
4624, 45syldan 282 . . . . . . . 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„‚)
47 eqid 2187 . . . . . . . . 9 (β„€β‰₯β€˜1) = (β„€β‰₯β€˜1)
48 simprl 529 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (β™―β€˜π΅) ∈ β„•)
49 nnuz 9576 . . . . . . . . . 10 β„• = (β„€β‰₯β€˜1)
5048, 49eleqtrdi 2280 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (β™―β€˜π΅) ∈ (β„€β‰₯β€˜1))
51 ssidd 3188 . . . . . . . . 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΅)))
5247, 50, 51fprodntrivap 11605 . . . . . . . 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 # 0 ∧ seqπ‘š( Β· , (𝑛 ∈ (β„€β‰₯β€˜1) ↦ if(𝑛 ∈ (1...(β™―β€˜π΅)), ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)), 1))) ⇝ 𝑦))
53 eleq1 2250 . . . . . . . . . . . . 13 (𝑗 = (π‘“β€˜π‘) β†’ (𝑗 ∈ 𝐴 ↔ (π‘“β€˜π‘) ∈ 𝐴))
5453dcbid 839 . . . . . . . . . . . 12 (𝑗 = (π‘“β€˜π‘) β†’ (DECID 𝑗 ∈ 𝐴 ↔ DECID (π‘“β€˜π‘) ∈ 𝐴))
5536ad3antrrr 492 . . . . . . . . . . . 12 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)
5613ad2antrr 488 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)
5756, 14syl 14 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ 𝑓:(1...(β™―β€˜π΅))⟢𝐡)
58 simpr 110 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ 𝑝 ∈ (1...(β™―β€˜π΅)))
5957, 58ffvelcdmd 5665 . . . . . . . . . . . 12 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ (π‘“β€˜π‘) ∈ 𝐡)
6054, 55, 59rspcdva 2858 . . . . . . . . . . 11 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ DECID (π‘“β€˜π‘) ∈ 𝐴)
61 f1ocnv 5486 . . . . . . . . . . . . . . 15 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ ◑𝑓:𝐡–1-1-ontoβ†’(1...(β™―β€˜π΅)))
62 f1of1 5472 . . . . . . . . . . . . . . 15 (◑𝑓:𝐡–1-1-ontoβ†’(1...(β™―β€˜π΅)) β†’ ◑𝑓:𝐡–1-1β†’(1...(β™―β€˜π΅)))
6356, 61, 623syl 17 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ ◑𝑓:𝐡–1-1β†’(1...(β™―β€˜π΅)))
641ad3antrrr 492 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ 𝐴 βŠ† 𝐡)
65 f1elima 5787 . . . . . . . . . . . . . 14 ((◑𝑓:𝐡–1-1β†’(1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘) ∈ 𝐡 ∧ 𝐴 βŠ† 𝐡) β†’ ((β—‘π‘“β€˜(π‘“β€˜π‘)) ∈ (◑𝑓 β€œ 𝐴) ↔ (π‘“β€˜π‘) ∈ 𝐴))
6663, 59, 64, 65syl3anc 1248 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ ((β—‘π‘“β€˜(π‘“β€˜π‘)) ∈ (◑𝑓 β€œ 𝐴) ↔ (π‘“β€˜π‘) ∈ 𝐴))
67 f1ocnvfv1 5791 . . . . . . . . . . . . . . 15 ((𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ (β—‘π‘“β€˜(π‘“β€˜π‘)) = 𝑝)
6856, 58, 67syl2anc 411 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ (β—‘π‘“β€˜(π‘“β€˜π‘)) = 𝑝)
6968eleq1d 2256 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ ((β—‘π‘“β€˜(π‘“β€˜π‘)) ∈ (◑𝑓 β€œ 𝐴) ↔ 𝑝 ∈ (◑𝑓 β€œ 𝐴)))
7066, 69bitr3d 190 . . . . . . . . . . . 12 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ ((π‘“β€˜π‘) ∈ 𝐴 ↔ 𝑝 ∈ (◑𝑓 β€œ 𝐴)))
7170dcbid 839 . . . . . . . . . . 11 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ (DECID (π‘“β€˜π‘) ∈ 𝐴 ↔ DECID 𝑝 ∈ (◑𝑓 β€œ 𝐴)))
7260, 71mpbid 147 . . . . . . . . . 10 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ DECID 𝑝 ∈ (◑𝑓 β€œ 𝐴))
7316ad2antrr 488 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ Β¬ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ (◑𝑓 β€œ 𝐴) βŠ† (1...(β™―β€˜π΅)))
74 simpr 110 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ Β¬ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ Β¬ 𝑝 ∈ (1...(β™―β€˜π΅)))
7573, 74ssneldd 3170 . . . . . . . . . . . 12 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ Β¬ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ Β¬ 𝑝 ∈ (◑𝑓 β€œ 𝐴))
7675olcd 735 . . . . . . . . . . 11 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ Β¬ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ (𝑝 ∈ (◑𝑓 β€œ 𝐴) ∨ Β¬ 𝑝 ∈ (◑𝑓 β€œ 𝐴)))
77 df-dc 836 . . . . . . . . . . 11 (DECID 𝑝 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑝 ∈ (◑𝑓 β€œ 𝐴) ∨ Β¬ 𝑝 ∈ (◑𝑓 β€œ 𝐴)))
7876, 77sylibr 134 . . . . . . . . . 10 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) ∧ Β¬ 𝑝 ∈ (1...(β™―β€˜π΅))) β†’ DECID 𝑝 ∈ (◑𝑓 β€œ 𝐴))
79 eluzelz 9550 . . . . . . . . . . . . 13 (𝑝 ∈ (β„€β‰₯β€˜1) β†’ 𝑝 ∈ β„€)
8079adantl 277 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) β†’ 𝑝 ∈ β„€)
81 1zzd 9293 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) β†’ 1 ∈ β„€)
8248adantr 276 . . . . . . . . . . . . 13 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) β†’ (β™―β€˜π΅) ∈ β„•)
8382nnzd 9387 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) β†’ (β™―β€˜π΅) ∈ β„€)
84 fzdcel 10053 . . . . . . . . . . . 12 ((𝑝 ∈ β„€ ∧ 1 ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„€) β†’ DECID 𝑝 ∈ (1...(β™―β€˜π΅)))
8580, 81, 83, 84syl3anc 1248 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) β†’ DECID 𝑝 ∈ (1...(β™―β€˜π΅)))
86 exmiddc 837 . . . . . . . . . . 11 (DECID 𝑝 ∈ (1...(β™―β€˜π΅)) β†’ (𝑝 ∈ (1...(β™―β€˜π΅)) ∨ Β¬ 𝑝 ∈ (1...(β™―β€˜π΅))))
8785, 86syl 14 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) β†’ (𝑝 ∈ (1...(β™―β€˜π΅)) ∨ Β¬ 𝑝 ∈ (1...(β™―β€˜π΅))))
8872, 78, 87mpjaodan 799 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑝 ∈ (β„€β‰₯β€˜1)) β†’ DECID 𝑝 ∈ (◑𝑓 β€œ 𝐴))
8988ralrimiva 2560 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆ€π‘ ∈ (β„€β‰₯β€˜1)DECID 𝑝 ∈ (◑𝑓 β€œ 𝐴))
90 1zzd 9293 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 1 ∈ β„€)
91 eldifi 3269 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ 𝑛 ∈ (1...(β™―β€˜π΅)))
9291, 20sylan2 286 . . . . . . . . . . 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)
93 eldifn 3270 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ Β¬ 𝑛 ∈ (◑𝑓 β€œ 𝐴))
9493adantl 277 . . . . . . . . . . . 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))
9591adantl 277 . . . . . . . . . . . . 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΅)))
9619adantr 276 . . . . . . . . . . . . 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)))
9795, 96mpbirand 441 . . . . . . . . . . . 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))
9894, 97mtbid 673 . . . . . . . . . . 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)
9992, 98eldifd 3151 . . . . . . . . . 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))
100 difss 3273 . . . . . . . . . . . . 13 (𝐡 βˆ– 𝐴) βŠ† 𝐡
101 resmpt 4967 . . . . . . . . . . . . 13 ((𝐡 βˆ– 𝐴) βŠ† 𝐡 β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴)) = (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢))
102100, 101ax-mp 5 . . . . . . . . . . . 12 ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴)) = (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)
103102fveq1i 5528 . . . . . . . . . . 11 (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴))β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›))
104 fvres 5551 . . . . . . . . . . 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‘›)))
105103, 104eqtr3id 2234 . . . . . . . . . 10 ((π‘“β€˜π‘›) ∈ (𝐡 βˆ– 𝐴) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
10699, 105syl 14 . . . . . . . . 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‘›)))
107 1ex 7965 . . . . . . . . . . . . . . 15 1 ∈ V
108107elsn2 3638 . . . . . . . . . . . . . 14 (𝐢 ∈ {1} ↔ 𝐢 = 1)
10929, 108sylibr 134 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ {1})
110109fmpttd 5684 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢):(𝐡 βˆ– 𝐴)⟢{1})
111110ad2antrr 488 . . . . . . . . . . 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})
112111, 99ffvelcdmd 5665 . . . . . . . . . 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})
113 elsni 3622 . . . . . . . . . 10 (((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ {1} β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = 1)
114112, 113syl 14 . . . . . . . . 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)
115106, 114eqtr3d 2222 . . . . . . . 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)
116 fzssuz 10078 . . . . . . . . 9 (1...(β™―β€˜π΅)) βŠ† (β„€β‰₯β€˜1)
117116a1i 9 . . . . . . . 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))
11885ralrimiva 2560 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆ€π‘ ∈ (β„€β‰₯β€˜1)DECID 𝑝 ∈ (1...(β™―β€˜π΅)))
11916, 46, 52, 89, 90, 115, 117, 118prodssdc 11610 . . . . . . 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‘›)))
1201adantr 276 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝐴 βŠ† 𝐡)
121120resmptd 4970 . . . . . . . . . . 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))
122121fveq1d 5529 . . . . . . . . . 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))
123 fvres 5551 . . . . . . . . . 10 (π‘š ∈ 𝐴 β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
124122, 123sylan9req 2241 . . . . . . . . 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))
125124prodeq2dv 11587 . . . . . . . 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))
126 fveq2 5527 . . . . . . . . 9 (π‘š = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
127 fprodss.4 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐡 ∈ Fin)
128127adantr 276 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝐡 ∈ Fin)
12936adantr 276 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)
130 ssfidc 6947 . . . . . . . . . . 11 ((𝐡 ∈ Fin ∧ 𝐴 βŠ† 𝐡 ∧ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴) β†’ 𝐴 ∈ Fin)
131128, 120, 129, 130syl3anc 1248 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝐴 ∈ Fin)
132120, 13, 131preimaf1ofi 6963 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (◑𝑓 β€œ 𝐴) ∈ Fin)
133 f1of1 5472 . . . . . . . . . . . 12 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))–1-1→𝐡)
13413, 133syl 14 . . . . . . . . . . 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)
135 f1ores 5488 . . . . . . . . . . 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)))
136134, 16, 135syl2anc 411 . . . . . . . . . 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)))
137 f1ofo 5480 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))–onto→𝐡)
13813, 137syl 14 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))–onto→𝐡)
139 foimacnv 5491 . . . . . . . . . . . 12 ((𝑓:(1...(β™―β€˜π΅))–onto→𝐡 ∧ 𝐴 βŠ† 𝐡) β†’ (𝑓 β€œ (◑𝑓 β€œ 𝐴)) = 𝐴)
140138, 120, 139syl2anc 411 . . . . . . . . . . 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)
141140f1oeq3d 5470 . . . . . . . . . 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))
142136, 141mpbid 147 . . . . . . . . 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)
143 fvres 5551 . . . . . . . . . 10 (𝑛 ∈ (◑𝑓 β€œ 𝐴) β†’ ((𝑓 β†Ύ (◑𝑓 β€œ 𝐴))β€˜π‘›) = (π‘“β€˜π‘›))
144143adantl 277 . . . . . . . . 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‘›))
145120sselda 3167 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ π‘š ∈ 𝐡)
14644ffvelcdmda 5664 . . . . . . . . . 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„‚)
147145, 146syldan 282 . . . . . . . . 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„‚)
148126, 132, 142, 144, 147fprodf1o 11609 . . . . . . . 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‘›)))
149125, 148eqtrd 2220 . . . . . . 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‘›)))
15048nnzd 9387 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (β™―β€˜π΅) ∈ β„€)
15190, 150fzfigd 10444 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (1...(β™―β€˜π΅)) ∈ Fin)
152 eqidd 2188 . . . . . . . 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‘›))
153126, 151, 13, 152, 146fprodf1o 11609 . . . . . . 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‘›)))
154119, 149, 1533eqtr4d 2230 . . . . . 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))
15525ralrimiva 2560 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘˜ ∈ 𝐴 𝐢 ∈ β„‚)
156155adantr 276 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆ€π‘˜ ∈ 𝐴 𝐢 ∈ β„‚)
157 prodfct 11608 . . . . . . 7 (βˆ€π‘˜ ∈ 𝐴 𝐢 ∈ β„‚ β†’ βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = βˆπ‘˜ ∈ 𝐴 𝐢)
158156, 157syl 14 . . . . . 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‘˜ ∈ 𝐴 𝐢)
15943ralrimiva 2560 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆ€π‘˜ ∈ 𝐡 𝐢 ∈ β„‚)
160 prodfct 11608 . . . . . . 7 (βˆ€π‘˜ ∈ 𝐡 𝐢 ∈ β„‚ β†’ βˆπ‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = βˆπ‘˜ ∈ 𝐡 𝐢)
161159, 160syl 14 . . . . . 6 ((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‘˜ ∈ 𝐡 𝐢)
162154, 158, 1613eqtr3d 2228 . . . . 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)
163162expr 375 . . . 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))
164163exlimdv 1829 . . 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))
165164expimpd 363 . 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))
166 fz1f1o 11396 . . 3 (𝐡 ∈ Fin β†’ (𝐡 = βˆ… ∨ ((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)))
167127, 166syl 14 . 2 (πœ‘ β†’ (𝐡 = βˆ… ∨ ((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)))
16811, 165, 167mpjaod 719 1 (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
Colors of variables: wff set class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1363  βˆƒwex 1502   ∈ wcel 2158  βˆ€wral 2465   βˆ– cdif 3138   βŠ† wss 3141  βˆ…c0 3434  {csn 3604   ↦ cmpt 4076  β—‘ccnv 4637   β†Ύ cres 4640   β€œ cima 4641   Fn wfn 5223  βŸΆwf 5224  β€“1-1β†’wf1 5225  β€“ontoβ†’wfo 5226  β€“1-1-ontoβ†’wf1o 5227  β€˜cfv 5228  (class class class)co 5888  Fincfn 6753  β„‚cc 7822  1c1 7825  β„•cn 8932  β„€cz 9266  β„€β‰₯cuz 9541  ...cfz 10021  β™―chash 10768  βˆcprod 11571
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-frec 6405  df-1o 6430  df-oadd 6434  df-er 6548  df-en 6754  df-dom 6755  df-fin 6756  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-fz 10022  df-fzo 10156  df-seqfrec 10459  df-exp 10533  df-ihash 10769  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-clim 11300  df-proddc 11572
This theorem is referenced by:  fprodsplitdc  11617
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