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Theorem breprexplema 33637
Description: Lemma for breprexp 33640 (induction step for weighted sums over representations). (Contributed by Thierry Arnoux, 7-Dec-2021.)
Hypotheses
Ref Expression
breprexp.n (πœ‘ β†’ 𝑁 ∈ β„•0)
breprexp.s (πœ‘ β†’ 𝑆 ∈ β„•0)
breprexplema.m (πœ‘ β†’ 𝑀 ∈ β„•0)
breprexplema.1 (πœ‘ β†’ 𝑀 ≀ ((𝑆 + 1) Β· 𝑁))
breprexplema.l (((πœ‘ ∧ π‘₯ ∈ (0..^(𝑆 + 1))) ∧ 𝑦 ∈ β„•) β†’ ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚)
Assertion
Ref Expression
breprexplema (πœ‘ β†’ Σ𝑑 ∈ ((1...𝑁)(reprβ€˜(𝑆 + 1))𝑀)βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))(βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)))
Distinct variable groups:   𝑆,π‘Ž   𝐿,π‘Ž,𝑏,𝑑,π‘₯,𝑦   𝑀,π‘Ž,𝑏,𝑑   𝑁,π‘Ž,𝑏,𝑑   𝑆,𝑏,𝑑,π‘₯,𝑦   πœ‘,π‘Ž,𝑏,𝑑,π‘₯,𝑦
Allowed substitution hints:   𝑀(π‘₯,𝑦)   𝑁(π‘₯,𝑦)
Proof of Theorem breprexplema
Dummy variables 𝑐 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fz1ssnn 13531 . . . . 5 (1...𝑁) βŠ† β„•
21a1i 11 . . . 4 (πœ‘ β†’ (1...𝑁) βŠ† β„•)
3 breprexplema.m . . . . 5 (πœ‘ β†’ 𝑀 ∈ β„•0)
43nn0zd 12583 . . . 4 (πœ‘ β†’ 𝑀 ∈ β„€)
5 breprexp.s . . . 4 (πœ‘ β†’ 𝑆 ∈ β„•0)
6 eqid 2732 . . . 4 (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) = (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
72, 4, 5, 6reprsuc 33622 . . 3 (πœ‘ β†’ ((1...𝑁)(reprβ€˜(𝑆 + 1))𝑀) = βˆͺ 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})))
87sumeq1d 15646 . 2 (πœ‘ β†’ Σ𝑑 ∈ ((1...𝑁)(reprβ€˜(𝑆 + 1))𝑀)βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = Σ𝑑 ∈ βˆͺ 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)))
9 fzfid 13937 . . 3 (πœ‘ β†’ (1...𝑁) ∈ Fin)
101a1i 11 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ (1...𝑁) βŠ† β„•)
114adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ 𝑀 ∈ β„€)
12 fzssz 13502 . . . . . . . 8 (1...𝑁) βŠ† β„€
13 simpr 485 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ 𝑏 ∈ (1...𝑁))
1412, 13sselid 3980 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ 𝑏 ∈ β„€)
1511, 14zsubcld 12670 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ (𝑀 βˆ’ 𝑏) ∈ β„€)
165adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ 𝑆 ∈ β„•0)
179adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ (1...𝑁) ∈ Fin)
1810, 15, 16, 17reprfi 33623 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ∈ Fin)
19 mptfi 9350 . . . . 5 (((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ∈ Fin β†’ (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) ∈ Fin)
2018, 19syl 17 . . . 4 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) ∈ Fin)
21 rnfi 9334 . . . 4 ((𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) ∈ Fin β†’ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) ∈ Fin)
2220, 21syl 17 . . 3 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) ∈ Fin)
2310, 15, 16reprval 33617 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) = {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = (𝑀 βˆ’ 𝑏)})
24 ssrab2 4077 . . . . 5 {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Ξ£π‘Ž ∈ (0..^𝑆)(π‘β€˜π‘Ž) = (𝑀 βˆ’ 𝑏)} βŠ† ((1...𝑁) ↑m (0..^𝑆))
2523, 24eqsstrdi 4036 . . . 4 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) βŠ† ((1...𝑁) ↑m (0..^𝑆)))
269elexd 3494 . . . 4 (πœ‘ β†’ (1...𝑁) ∈ V)
27 fzonel 13645 . . . . 5 Β¬ 𝑆 ∈ (0..^𝑆)
2827a1i 11 . . . 4 (πœ‘ β†’ Β¬ 𝑆 ∈ (0..^𝑆))
2925, 26, 5, 28, 6actfunsnrndisj 33612 . . 3 (πœ‘ β†’ Disj 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})))
30 fzofi 13938 . . . . . 6 (0..^(𝑆 + 1)) ∈ Fin
3130a1i 11 . . . . 5 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) β†’ (0..^(𝑆 + 1)) ∈ Fin)
32 breprexplema.l . . . . . . . . 9 (((πœ‘ ∧ π‘₯ ∈ (0..^(𝑆 + 1))) ∧ 𝑦 ∈ β„•) β†’ ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚)
3332ralrimiva 3146 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ (0..^(𝑆 + 1))) β†’ βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚)
3433ralrimiva 3146 . . . . . . 7 (πœ‘ β†’ βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚)
3534ad3antrrr 728 . . . . . 6 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚)
36 simpr 485 . . . . . . 7 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ π‘Ž ∈ (0..^(𝑆 + 1)))
37 nfv 1917 . . . . . . . . . . . 12 Ⅎ𝑣(πœ‘ ∧ 𝑏 ∈ (1...𝑁))
38 nfcv 2903 . . . . . . . . . . . . 13 Ⅎ𝑣𝑑
39 nfmpt1 5256 . . . . . . . . . . . . . 14 Ⅎ𝑣(𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
4039nfrn 5951 . . . . . . . . . . . . 13 Ⅎ𝑣ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
4138, 40nfel 2917 . . . . . . . . . . . 12 Ⅎ𝑣 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
4237, 41nfan 1902 . . . . . . . . . . 11 Ⅎ𝑣((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})))
431a1i 11 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ (1...𝑁) βŠ† β„•)
4415ad3antrrr 728 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ (𝑀 βˆ’ 𝑏) ∈ β„€)
4516ad3antrrr 728 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ 𝑆 ∈ β„•0)
46 simplr 767 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)))
4743, 44, 45, 46reprf 33619 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ 𝑣:(0..^𝑆)⟢(1...𝑁))
4813ad3antrrr 728 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ 𝑏 ∈ (1...𝑁))
4945, 48fsnd 6876 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ {βŸ¨π‘†, π‘βŸ©}:{𝑆}⟢(1...𝑁))
50 fzodisjsn 13669 . . . . . . . . . . . . . 14 ((0..^𝑆) ∩ {𝑆}) = βˆ…
5150a1i 11 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ ((0..^𝑆) ∩ {𝑆}) = βˆ…)
52 fun2 6754 . . . . . . . . . . . . 13 (((𝑣:(0..^𝑆)⟢(1...𝑁) ∧ {βŸ¨π‘†, π‘βŸ©}:{𝑆}⟢(1...𝑁)) ∧ ((0..^𝑆) ∩ {𝑆}) = βˆ…) β†’ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}):((0..^𝑆) βˆͺ {𝑆})⟢(1...𝑁))
5347, 49, 51, 52syl21anc 836 . . . . . . . . . . . 12 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}):((0..^𝑆) βˆͺ {𝑆})⟢(1...𝑁))
54 simpr 485 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
55 nn0uz 12863 . . . . . . . . . . . . . . . 16 β„•0 = (β„€β‰₯β€˜0)
565, 55eleqtrdi 2843 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑆 ∈ (β„€β‰₯β€˜0))
57 fzosplitsn 13739 . . . . . . . . . . . . . . 15 (𝑆 ∈ (β„€β‰₯β€˜0) β†’ (0..^(𝑆 + 1)) = ((0..^𝑆) βˆͺ {𝑆}))
5856, 57syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ (0..^(𝑆 + 1)) = ((0..^𝑆) βˆͺ {𝑆}))
5958ad4antr 730 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ (0..^(𝑆 + 1)) = ((0..^𝑆) βˆͺ {𝑆}))
6054, 59feq12d 6705 . . . . . . . . . . . 12 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ (𝑑:(0..^(𝑆 + 1))⟢(1...𝑁) ↔ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}):((0..^𝑆) βˆͺ {𝑆})⟢(1...𝑁)))
6153, 60mpbird 256 . . . . . . . . . . 11 (((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ 𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) β†’ 𝑑:(0..^(𝑆 + 1))⟢(1...𝑁))
62 simpr 485 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) β†’ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})))
63 vex 3478 . . . . . . . . . . . . . 14 𝑣 ∈ V
64 snex 5431 . . . . . . . . . . . . . 14 {βŸ¨π‘†, π‘βŸ©} ∈ V
6563, 64unex 7732 . . . . . . . . . . . . 13 (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}) ∈ V
666, 65elrnmpti 5959 . . . . . . . . . . . 12 (𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) ↔ βˆƒπ‘£ ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
6762, 66sylib 217 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) β†’ βˆƒπ‘£ ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))𝑑 = (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
6842, 61, 67r19.29af 3265 . . . . . . . . . 10 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) β†’ 𝑑:(0..^(𝑆 + 1))⟢(1...𝑁))
6968adantr 481 . . . . . . . . 9 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ 𝑑:(0..^(𝑆 + 1))⟢(1...𝑁))
7069, 36ffvelcdmd 7087 . . . . . . . 8 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ (π‘‘β€˜π‘Ž) ∈ (1...𝑁))
711, 70sselid 3980 . . . . . . 7 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ (π‘‘β€˜π‘Ž) ∈ β„•)
72 fveq2 6891 . . . . . . . . . 10 (π‘₯ = π‘Ž β†’ (πΏβ€˜π‘₯) = (πΏβ€˜π‘Ž))
7372fveq1d 6893 . . . . . . . . 9 (π‘₯ = π‘Ž β†’ ((πΏβ€˜π‘₯)β€˜π‘¦) = ((πΏβ€˜π‘Ž)β€˜π‘¦))
7473eleq1d 2818 . . . . . . . 8 (π‘₯ = π‘Ž β†’ (((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚ ↔ ((πΏβ€˜π‘Ž)β€˜π‘¦) ∈ β„‚))
75 fveq2 6891 . . . . . . . . 9 (𝑦 = (π‘‘β€˜π‘Ž) β†’ ((πΏβ€˜π‘Ž)β€˜π‘¦) = ((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)))
7675eleq1d 2818 . . . . . . . 8 (𝑦 = (π‘‘β€˜π‘Ž) β†’ (((πΏβ€˜π‘Ž)β€˜π‘¦) ∈ β„‚ ↔ ((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) ∈ β„‚))
7774, 76rspc2v 3622 . . . . . . 7 ((π‘Ž ∈ (0..^(𝑆 + 1)) ∧ (π‘‘β€˜π‘Ž) ∈ β„•) β†’ (βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚ β†’ ((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) ∈ β„‚))
7836, 71, 77syl2anc 584 . . . . . 6 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ (βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚ β†’ ((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) ∈ β„‚))
7935, 78mpd 15 . . . . 5 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ ((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) ∈ β„‚)
8031, 79fprodcl 15895 . . . 4 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))) β†’ βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) ∈ β„‚)
8180anasss 467 . . 3 ((πœ‘ ∧ (𝑏 ∈ (1...𝑁) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})))) β†’ βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) ∈ β„‚)
829, 22, 29, 81fsumiun 15766 . 2 (πœ‘ β†’ Σ𝑑 ∈ βˆͺ 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)))
8358ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ (0..^(𝑆 + 1)) = ((0..^𝑆) βˆͺ {𝑆}))
8483prodeq1d 15864 . . . . . 6 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) = βˆπ‘Ž ∈ ((0..^𝑆) βˆͺ {𝑆})((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)))
85 nfv 1917 . . . . . . 7 β„²π‘Ž((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)))
86 nfcv 2903 . . . . . . 7 β„²π‘Ž((πΏβ€˜π‘†)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†))
87 fzofi 13938 . . . . . . . 8 (0..^𝑆) ∈ Fin
8887a1i 11 . . . . . . 7 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ (0..^𝑆) ∈ Fin)
8916adantr 481 . . . . . . 7 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ 𝑆 ∈ β„•0)
9027a1i 11 . . . . . . 7 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ Β¬ 𝑆 ∈ (0..^𝑆))
911a1i 11 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ (1...𝑁) βŠ† β„•)
9215adantr 481 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ (𝑀 βˆ’ 𝑏) ∈ β„€)
93 simpr 485 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)))
9491, 92, 89, 93reprf 33619 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ 𝑒:(0..^𝑆)⟢(1...𝑁))
9594ffnd 6718 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ 𝑒 Fn (0..^𝑆))
9695adantr 481 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ 𝑒 Fn (0..^𝑆))
9713adantr 481 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ 𝑏 ∈ (1...𝑁))
98 fnsng 6600 . . . . . . . . . . . 12 ((𝑆 ∈ β„•0 ∧ 𝑏 ∈ (1...𝑁)) β†’ {βŸ¨π‘†, π‘βŸ©} Fn {𝑆})
9989, 97, 98syl2anc 584 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ {βŸ¨π‘†, π‘βŸ©} Fn {𝑆})
10099adantr 481 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ {βŸ¨π‘†, π‘βŸ©} Fn {𝑆})
10150a1i 11 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ ((0..^𝑆) ∩ {𝑆}) = βˆ…)
102 simpr 485 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ π‘Ž ∈ (0..^𝑆))
103 fvun1 6982 . . . . . . . . . 10 ((𝑒 Fn (0..^𝑆) ∧ {βŸ¨π‘†, π‘βŸ©} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = βˆ… ∧ π‘Ž ∈ (0..^𝑆))) β†’ ((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž) = (π‘’β€˜π‘Ž))
10496, 100, 101, 102, 103syl112anc 1374 . . . . . . . . 9 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ ((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž) = (π‘’β€˜π‘Ž))
105104fveq2d 6895 . . . . . . . 8 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ ((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) = ((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)))
10634ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚)
107106adantr 481 . . . . . . . . 9 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚)
108 fzossfzop1 13709 . . . . . . . . . . . . 13 (𝑆 ∈ β„•0 β†’ (0..^𝑆) βŠ† (0..^(𝑆 + 1)))
1095, 108syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (0..^𝑆) βŠ† (0..^(𝑆 + 1)))
110109ad2antrr 724 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ (0..^𝑆) βŠ† (0..^(𝑆 + 1)))
111110sselda 3982 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ π‘Ž ∈ (0..^(𝑆 + 1)))
11294ffvelcdmda 7086 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ (π‘’β€˜π‘Ž) ∈ (1...𝑁))
1131, 112sselid 3980 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ (π‘’β€˜π‘Ž) ∈ β„•)
114 fveq2 6891 . . . . . . . . . . . 12 (𝑦 = (π‘’β€˜π‘Ž) β†’ ((πΏβ€˜π‘Ž)β€˜π‘¦) = ((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)))
115114eleq1d 2818 . . . . . . . . . . 11 (𝑦 = (π‘’β€˜π‘Ž) β†’ (((πΏβ€˜π‘Ž)β€˜π‘¦) ∈ β„‚ ↔ ((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) ∈ β„‚))
11674, 115rspc2v 3622 . . . . . . . . . 10 ((π‘Ž ∈ (0..^(𝑆 + 1)) ∧ (π‘’β€˜π‘Ž) ∈ β„•) β†’ (βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚ β†’ ((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) ∈ β„‚))
117111, 113, 116syl2anc 584 . . . . . . . . 9 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ (βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚ β†’ ((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) ∈ β„‚))
118107, 117mpd 15 . . . . . . . 8 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ ((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) ∈ β„‚)
119105, 118eqeltrd 2833 . . . . . . 7 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ ((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) ∈ β„‚)
120 fveq2 6891 . . . . . . . 8 (π‘Ž = 𝑆 β†’ (πΏβ€˜π‘Ž) = (πΏβ€˜π‘†))
121 fveq2 6891 . . . . . . . 8 (π‘Ž = 𝑆 β†’ ((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž) = ((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†))
122120, 121fveq12d 6898 . . . . . . 7 (π‘Ž = 𝑆 β†’ ((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) = ((πΏβ€˜π‘†)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†)))
12350a1i 11 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ ((0..^𝑆) ∩ {𝑆}) = βˆ…)
124 snidg 4662 . . . . . . . . . . . 12 (𝑆 ∈ β„•0 β†’ 𝑆 ∈ {𝑆})
12589, 124syl 17 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ 𝑆 ∈ {𝑆})
126 fvun2 6983 . . . . . . . . . . 11 ((𝑒 Fn (0..^𝑆) ∧ {βŸ¨π‘†, π‘βŸ©} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = βˆ… ∧ 𝑆 ∈ {𝑆})) β†’ ((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†) = ({βŸ¨π‘†, π‘βŸ©}β€˜π‘†))
12795, 99, 123, 125, 126syl112anc 1374 . . . . . . . . . 10 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ ((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†) = ({βŸ¨π‘†, π‘βŸ©}β€˜π‘†))
128 fvsng 7177 . . . . . . . . . . 11 ((𝑆 ∈ β„•0 ∧ 𝑏 ∈ (1...𝑁)) β†’ ({βŸ¨π‘†, π‘βŸ©}β€˜π‘†) = 𝑏)
12989, 97, 128syl2anc 584 . . . . . . . . . 10 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ ({βŸ¨π‘†, π‘βŸ©}β€˜π‘†) = 𝑏)
130127, 129eqtrd 2772 . . . . . . . . 9 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ ((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†) = 𝑏)
131130fveq2d 6895 . . . . . . . 8 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ ((πΏβ€˜π‘†)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†)) = ((πΏβ€˜π‘†)β€˜π‘))
132 fzonn0p1 13708 . . . . . . . . . . . 12 (𝑆 ∈ β„•0 β†’ 𝑆 ∈ (0..^(𝑆 + 1)))
1335, 132syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝑆 ∈ (0..^(𝑆 + 1)))
134133ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ 𝑆 ∈ (0..^(𝑆 + 1)))
1351, 97sselid 3980 . . . . . . . . . 10 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ 𝑏 ∈ β„•)
136 fveq2 6891 . . . . . . . . . . . . 13 (π‘₯ = 𝑆 β†’ (πΏβ€˜π‘₯) = (πΏβ€˜π‘†))
137136fveq1d 6893 . . . . . . . . . . . 12 (π‘₯ = 𝑆 β†’ ((πΏβ€˜π‘₯)β€˜π‘¦) = ((πΏβ€˜π‘†)β€˜π‘¦))
138137eleq1d 2818 . . . . . . . . . . 11 (π‘₯ = 𝑆 β†’ (((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚ ↔ ((πΏβ€˜π‘†)β€˜π‘¦) ∈ β„‚))
139 fveq2 6891 . . . . . . . . . . . 12 (𝑦 = 𝑏 β†’ ((πΏβ€˜π‘†)β€˜π‘¦) = ((πΏβ€˜π‘†)β€˜π‘))
140139eleq1d 2818 . . . . . . . . . . 11 (𝑦 = 𝑏 β†’ (((πΏβ€˜π‘†)β€˜π‘¦) ∈ β„‚ ↔ ((πΏβ€˜π‘†)β€˜π‘) ∈ β„‚))
141138, 140rspc2v 3622 . . . . . . . . . 10 ((𝑆 ∈ (0..^(𝑆 + 1)) ∧ 𝑏 ∈ β„•) β†’ (βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚ β†’ ((πΏβ€˜π‘†)β€˜π‘) ∈ β„‚))
142134, 135, 141syl2anc 584 . . . . . . . . 9 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ (βˆ€π‘₯ ∈ (0..^(𝑆 + 1))βˆ€π‘¦ ∈ β„• ((πΏβ€˜π‘₯)β€˜π‘¦) ∈ β„‚ β†’ ((πΏβ€˜π‘†)β€˜π‘) ∈ β„‚))
143106, 142mpd 15 . . . . . . . 8 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ ((πΏβ€˜π‘†)β€˜π‘) ∈ β„‚)
144131, 143eqeltrd 2833 . . . . . . 7 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ ((πΏβ€˜π‘†)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†)) ∈ β„‚)
14585, 86, 88, 89, 90, 119, 122, 144fprodsplitsn 15932 . . . . . 6 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ βˆπ‘Ž ∈ ((0..^𝑆) βˆͺ {𝑆})((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) = (βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†))))
146105prodeq2dv 15866 . . . . . . 7 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) = βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)))
147146, 131oveq12d 7426 . . . . . 6 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ (βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘†))) = (βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)))
14884, 145, 1473eqtrd 2776 . . . . 5 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) = (βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)))
149148sumeq2dv 15648 . . . 4 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ Σ𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)) = Σ𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))(βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)))
150 simpl 483 . . . . . . . 8 ((𝑑 = (𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©}) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ 𝑑 = (𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
151150fveq1d 6893 . . . . . . 7 ((𝑑 = (𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©}) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ (π‘‘β€˜π‘Ž) = ((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž))
152151fveq2d 6895 . . . . . 6 ((𝑑 = (𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©}) ∧ π‘Ž ∈ (0..^(𝑆 + 1))) β†’ ((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = ((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)))
153152prodeq2dv 15866 . . . . 5 (𝑑 = (𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©}) β†’ βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)))
15425, 26, 5, 28, 6actfunsnf1o 33611 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})):((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))–1-1-ontoβ†’ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})))
1556a1i 11 . . . . . 6 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})) = (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©})))
156 simpr 485 . . . . . . 7 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑣 = 𝑒) β†’ 𝑣 = 𝑒)
157156uneq1d 4162 . . . . . 6 ((((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) ∧ 𝑣 = 𝑒) β†’ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}) = (𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
158 vex 3478 . . . . . . . 8 𝑒 ∈ V
159158, 64unex 7732 . . . . . . 7 (𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©}) ∈ V
160159a1i 11 . . . . . 6 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ (𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©}) ∈ V)
161155, 157, 93, 160fvmptd 7005 . . . . 5 (((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))) β†’ ((𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))β€˜π‘’) = (𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©}))
162153, 18, 154, 161, 80fsumf1o 15668 . . . 4 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = Σ𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜((𝑒 βˆͺ {βŸ¨π‘†, π‘βŸ©})β€˜π‘Ž)))
163 simpl 483 . . . . . . . . . 10 ((𝑑 = 𝑒 ∧ π‘Ž ∈ (0..^𝑆)) β†’ 𝑑 = 𝑒)
164163fveq1d 6893 . . . . . . . . 9 ((𝑑 = 𝑒 ∧ π‘Ž ∈ (0..^𝑆)) β†’ (π‘‘β€˜π‘Ž) = (π‘’β€˜π‘Ž))
165164fveq2d 6895 . . . . . . . 8 ((𝑑 = 𝑒 ∧ π‘Ž ∈ (0..^𝑆)) β†’ ((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = ((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)))
166165prodeq2dv 15866 . . . . . . 7 (𝑑 = 𝑒 β†’ βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)))
167166oveq1d 7423 . . . . . 6 (𝑑 = 𝑒 β†’ (βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)) = (βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)))
168167cbvsumv 15641 . . . . 5 Σ𝑑 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))(βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)) = Σ𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))(βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘))
169168a1i 11 . . . 4 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ Σ𝑑 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))(βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)) = Σ𝑒 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))(βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘’β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)))
170149, 162, 1693eqtr4d 2782 . . 3 ((πœ‘ ∧ 𝑏 ∈ (1...𝑁)) β†’ Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = Σ𝑑 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))(βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)))
171170sumeq2dv 15648 . 2 (πœ‘ β†’ Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏)) ↦ (𝑣 βˆͺ {βŸ¨π‘†, π‘βŸ©}))βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))(βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)))
1728, 82, 1713eqtrd 2776 1 (πœ‘ β†’ Σ𝑑 ∈ ((1...𝑁)(reprβ€˜(𝑆 + 1))𝑀)βˆπ‘Ž ∈ (0..^(𝑆 + 1))((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(reprβ€˜π‘†)(𝑀 βˆ’ 𝑏))(βˆπ‘Ž ∈ (0..^𝑆)((πΏβ€˜π‘Ž)β€˜(π‘‘β€˜π‘Ž)) Β· ((πΏβ€˜π‘†)β€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βŸ¨cop 4634  βˆͺ ciun 4997   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819  Fincfn 8938  β„‚cc 11107  0cc0 11109  1c1 11110   + caddc 11112   Β· cmul 11114   ≀ cle 11248   βˆ’ cmin 11443  β„•cn 12211  β„•0cn0 12471  β„€cz 12557  β„€β‰₯cuz 12821  ...cfz 13483  ..^cfzo 13626  Ξ£csu 15631  βˆcprod 15848  reprcrepr 33615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-oi 9504  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-n0 12472  df-z 12558  df-uz 12822  df-rp 12974  df-fz 13484  df-fzo 13627  df-seq 13966  df-exp 14027  df-hash 14290  df-cj 15045  df-re 15046  df-im 15047  df-sqrt 15181  df-abs 15182  df-clim 15431  df-sum 15632  df-prod 15849  df-repr 33616
This theorem is referenced by:  breprexplemc  33639
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