MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscatd2 Structured version   Visualization version   GIF version

Theorem iscatd2 17726
Description: Version of iscatd 17718 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
iscatd2.b (𝜑𝐵 = (Base‘𝐶))
iscatd2.h (𝜑𝐻 = (Hom ‘𝐶))
iscatd2.o (𝜑· = (comp‘𝐶))
iscatd2.c (𝜑𝐶𝑉)
iscatd2.ps (𝜓 ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
iscatd2.1 ((𝜑𝑦𝐵) → 1 ∈ (𝑦𝐻𝑦))
iscatd2.2 ((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
iscatd2.3 ((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
iscatd2.4 ((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
iscatd2.5 ((𝜑𝜓) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
Assertion
Ref Expression
iscatd2 (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦𝐵1 )))
Distinct variable groups:   𝑓,𝑔,𝑘,𝑤,𝑥,𝑧, 1   𝑦,𝑓,𝐵,𝑔,𝑘,𝑤,𝑥,𝑧   𝐶,𝑔,𝑘,𝑤,𝑦,𝑧   𝑓,𝐻,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧   𝜑,𝑓,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧   · ,𝑓,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,𝑘)   𝐶(𝑥,𝑓)   1 (𝑦)   𝑉(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,𝑘)

Proof of Theorem iscatd2
Dummy variables 𝑎 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscatd2.b . . 3 (𝜑𝐵 = (Base‘𝐶))
2 iscatd2.h . . 3 (𝜑𝐻 = (Hom ‘𝐶))
3 iscatd2.o . . 3 (𝜑· = (comp‘𝐶))
4 iscatd2.c . . 3 (𝜑𝐶𝑉)
5 iscatd2.1 . . 3 ((𝜑𝑦𝐵) → 1 ∈ (𝑦𝐻𝑦))
65ne0d 4348 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦𝐻𝑦) ≠ ∅)
763ad2antr1 1187 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → (𝑦𝐻𝑦) ≠ ∅)
8 n0 4359 . . . . 5 ((𝑦𝐻𝑦) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑦𝐻𝑦))
97, 8sylib 218 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ∃𝑔 𝑔 ∈ (𝑦𝐻𝑦))
10 n0 4359 . . . . 5 ((𝑦𝐻𝑦) ≠ ∅ ↔ ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦))
117, 10sylib 218 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦))
12 exdistrv 1953 . . . . 5 (∃𝑔𝑘(𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)) ↔ (∃𝑔 𝑔 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦)))
13 simpll 767 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝜑)
14 simplr2 1215 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑎𝐵)
15 simplr1 1214 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑦𝐵)
1614, 15jca 511 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → (𝑎𝐵𝑦𝐵))
17 simplr3 1216 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑟 ∈ (𝑎𝐻𝑦))
18 simprl 771 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑔 ∈ (𝑦𝐻𝑦))
19 simprr 773 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑘 ∈ (𝑦𝐻𝑦))
2017, 18, 193jca 1127 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))
21 iscatd2.ps . . . . . . . . . . . . . . 15 (𝜓 ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
22 simplll 775 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑥 = 𝑎)
2322eleq1d 2824 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑥𝐵𝑎𝐵))
2423anbi1d 631 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ (𝑎𝐵𝑦𝐵)))
25 simpllr 776 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑧 = 𝑦)
2625eleq1d 2824 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑧𝐵𝑦𝐵))
27 simplr 769 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑤 = 𝑦)
2827eleq1d 2824 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑤𝐵𝑦𝐵))
2926, 28anbi12d 632 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ (𝑦𝐵𝑦𝐵)))
30 anidm 564 . . . . . . . . . . . . . . . . 17 ((𝑦𝐵𝑦𝐵) ↔ 𝑦𝐵)
3129, 30bitrdi 287 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ 𝑦𝐵))
32 simpr 484 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑓 = 𝑟)
3322oveq1d 7446 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑥𝐻𝑦) = (𝑎𝐻𝑦))
3432, 33eleq12d 2833 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑟 ∈ (𝑎𝐻𝑦)))
3525oveq2d 7447 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑦𝐻𝑧) = (𝑦𝐻𝑦))
3635eleq2d 2825 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑔 ∈ (𝑦𝐻𝑦)))
3725, 27oveq12d 7449 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑧𝐻𝑤) = (𝑦𝐻𝑦))
3837eleq2d 2825 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑘 ∈ (𝑧𝐻𝑤) ↔ 𝑘 ∈ (𝑦𝐻𝑦)))
3934, 36, 383anbi123d 1435 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))))
4024, 31, 393anbi123d 1435 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))))
4121, 40bitrid 283 . . . . . . . . . . . . . 14 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝜓 ↔ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))))
4241anbi2d 630 . . . . . . . . . . . . 13 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))))))
4322opeq1d 4884 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑦⟩)
4443oveq1d 7446 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (⟨𝑥, 𝑦· 𝑦) = (⟨𝑎, 𝑦· 𝑦))
45 eqidd 2736 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 1 = 1 )
4644, 45, 32oveq123d 7452 . . . . . . . . . . . . . 14 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟))
4746, 32eqeq12d 2751 . . . . . . . . . . . . 13 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓 ↔ ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟))
4842, 47imbi12d 344 . . . . . . . . . . . 12 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓) ↔ ((𝜑 ∧ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)))
4948sbiedvw 2093 . . . . . . . . . . 11 (((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) → ([𝑟 / 𝑓]((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓) ↔ ((𝜑 ∧ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)))
5049sbiedvw 2093 . . . . . . . . . 10 ((𝑥 = 𝑎𝑧 = 𝑦) → ([𝑦 / 𝑤][𝑟 / 𝑓]((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓) ↔ ((𝜑 ∧ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)))
5150sbiedvw 2093 . . . . . . . . 9 (𝑥 = 𝑎 → ([𝑦 / 𝑧][𝑦 / 𝑤][𝑟 / 𝑓]((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓) ↔ ((𝜑 ∧ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)))
52 iscatd2.2 . . . . . . . . . . . 12 ((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
5352sbt 2064 . . . . . . . . . . 11 [𝑟 / 𝑓]((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
5453sbt 2064 . . . . . . . . . 10 [𝑦 / 𝑤][𝑟 / 𝑓]((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
5554sbt 2064 . . . . . . . . 9 [𝑦 / 𝑧][𝑦 / 𝑤][𝑟 / 𝑓]((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
5651, 55chvarvv 1996 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)
5713, 16, 15, 20, 56syl13anc 1371 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)
5857ex 412 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ((𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟))
5958exlimdvv 1932 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → (∃𝑔𝑘(𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟))
6012, 59biimtrrid 243 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ((∃𝑔 𝑔 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦)) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟))
619, 11, 60mp2and 699 . . 3 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)
6263ad2antr1 1187 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → (𝑦𝐻𝑦) ≠ ∅)
63 n0 4359 . . . . 5 ((𝑦𝐻𝑦) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑦𝐻𝑦))
6462, 63sylib 218 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ∃𝑓 𝑓 ∈ (𝑦𝐻𝑦))
65 id 22 . . . . . . . 8 (𝑦 = 𝑎𝑦 = 𝑎)
6665, 65oveq12d 7449 . . . . . . 7 (𝑦 = 𝑎 → (𝑦𝐻𝑦) = (𝑎𝐻𝑎))
6766neeq1d 2998 . . . . . 6 (𝑦 = 𝑎 → ((𝑦𝐻𝑦) ≠ ∅ ↔ (𝑎𝐻𝑎) ≠ ∅))
686ralrimiva 3144 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑦𝐻𝑦) ≠ ∅)
6968adantr 480 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ∀𝑦𝐵 (𝑦𝐻𝑦) ≠ ∅)
70 simpr2 1194 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → 𝑎𝐵)
7167, 69, 70rspcdva 3623 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → (𝑎𝐻𝑎) ≠ ∅)
72 n0 4359 . . . . 5 ((𝑎𝐻𝑎) ≠ ∅ ↔ ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎))
7371, 72sylib 218 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎))
74 exdistrv 1953 . . . . 5 (∃𝑓𝑘(𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎)) ↔ (∃𝑓 𝑓 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎)))
75 simpll 767 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝜑)
76 simplr1 1214 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑦𝐵)
77 simplr2 1215 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑎𝐵)
78 simprl 771 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑓 ∈ (𝑦𝐻𝑦))
79 simplr3 1216 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑟 ∈ (𝑦𝐻𝑎))
80 simprr 773 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑘 ∈ (𝑎𝐻𝑎))
8178, 79, 803jca 1127 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))
82 simplll 775 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑥 = 𝑦)
8382eleq1d 2824 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑥𝐵𝑦𝐵))
8483anbi1d 631 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ (𝑦𝐵𝑦𝐵)))
8584, 30bitrdi 287 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ 𝑦𝐵))
86 simpllr 776 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑧 = 𝑎)
8786eleq1d 2824 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑧𝐵𝑎𝐵))
88 simplr 769 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑤 = 𝑎)
8988eleq1d 2824 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑤𝐵𝑎𝐵))
9087, 89anbi12d 632 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ (𝑎𝐵𝑎𝐵)))
91 anidm 564 . . . . . . . . . . . . . . . . 17 ((𝑎𝐵𝑎𝐵) ↔ 𝑎𝐵)
9290, 91bitrdi 287 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ 𝑎𝐵))
9382oveq1d 7446 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑥𝐻𝑦) = (𝑦𝐻𝑦))
9493eleq2d 2825 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑦𝐻𝑦)))
95 simpr 484 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑔 = 𝑟)
9686oveq2d 7447 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑦𝐻𝑧) = (𝑦𝐻𝑎))
9795, 96eleq12d 2833 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑟 ∈ (𝑦𝐻𝑎)))
9886, 88oveq12d 7449 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑧𝐻𝑤) = (𝑎𝐻𝑎))
9998eleq2d 2825 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑘 ∈ (𝑧𝐻𝑤) ↔ 𝑘 ∈ (𝑎𝐻𝑎)))
10094, 97, 993anbi123d 1435 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎))))
10185, 92, 1003anbi123d 1435 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))))
10221, 101bitrid 283 . . . . . . . . . . . . . 14 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝜓 ↔ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))))
103102anbi2d 630 . . . . . . . . . . . . 13 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎))))))
10486oveq2d 7447 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (⟨𝑦, 𝑦· 𝑧) = (⟨𝑦, 𝑦· 𝑎))
105 eqidd 2736 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 1 = 1 )
106104, 95, 105oveq123d 7452 . . . . . . . . . . . . . 14 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ))
107106, 95eqeq12d 2751 . . . . . . . . . . . . 13 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔 ↔ (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟))
108103, 107imbi12d 344 . . . . . . . . . . . 12 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔) ↔ ((𝜑 ∧ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)))
109108sbiedvw 2093 . . . . . . . . . . 11 (((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) → ([𝑟 / 𝑔]((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔) ↔ ((𝜑 ∧ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)))
110109sbiedvw 2093 . . . . . . . . . 10 ((𝑥 = 𝑦𝑧 = 𝑎) → ([𝑎 / 𝑤][𝑟 / 𝑔]((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔) ↔ ((𝜑 ∧ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)))
111110sbiedvw 2093 . . . . . . . . 9 (𝑥 = 𝑦 → ([𝑎 / 𝑧][𝑎 / 𝑤][𝑟 / 𝑔]((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔) ↔ ((𝜑 ∧ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)))
112 iscatd2.3 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
113112sbt 2064 . . . . . . . . . . 11 [𝑟 / 𝑔]((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
114113sbt 2064 . . . . . . . . . 10 [𝑎 / 𝑤][𝑟 / 𝑔]((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
115114sbt 2064 . . . . . . . . 9 [𝑎 / 𝑧][𝑎 / 𝑤][𝑟 / 𝑔]((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
116111, 115chvarvv 1996 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)
11775, 76, 77, 81, 116syl13anc 1371 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)
118117ex 412 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ((𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎)) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟))
119118exlimdvv 1932 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → (∃𝑓𝑘(𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎)) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟))
12074, 119biimtrrid 243 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ((∃𝑓 𝑓 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎)) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟))
12164, 73, 120mp2and 699 . . 3 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)
122 id 22 . . . . . . . 8 (𝑦 = 𝑧𝑦 = 𝑧)
123122, 122oveq12d 7449 . . . . . . 7 (𝑦 = 𝑧 → (𝑦𝐻𝑦) = (𝑧𝐻𝑧))
124123neeq1d 2998 . . . . . 6 (𝑦 = 𝑧 → ((𝑦𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑧) ≠ ∅))
125683ad2ant1 1132 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → ∀𝑦𝐵 (𝑦𝐻𝑦) ≠ ∅)
126 simp23 1207 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → 𝑧𝐵)
127124, 125, 126rspcdva 3623 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (𝑧𝐻𝑧) ≠ ∅)
128 n0 4359 . . . . 5 ((𝑧𝐻𝑧) ≠ ∅ ↔ ∃𝑘 𝑘 ∈ (𝑧𝐻𝑧))
129127, 128sylib 218 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → ∃𝑘 𝑘 ∈ (𝑧𝐻𝑧))
130 eleq1w 2822 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
1311303anbi1d 1439 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥𝐵𝑎𝐵𝑧𝐵) ↔ (𝑦𝐵𝑎𝐵𝑧𝐵)))
132 oveq1 7438 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑥𝐻𝑎) = (𝑦𝐻𝑎))
133132eleq2d 2825 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑟 ∈ (𝑥𝐻𝑎) ↔ 𝑟 ∈ (𝑦𝐻𝑎)))
134133anbi1d 631 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ↔ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))))
135134anbi1d 631 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)) ↔ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))
136131, 135anbi12d 632 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))) ↔ ((𝑦𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))))
137136anbi2d 630 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) ↔ (𝜑 ∧ ((𝑦𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))))
138 opeq1 4878 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ⟨𝑥, 𝑎⟩ = ⟨𝑦, 𝑎⟩)
139138oveq1d 7446 . . . . . . . . . . 11 (𝑥 = 𝑦 → (⟨𝑥, 𝑎· 𝑧) = (⟨𝑦, 𝑎· 𝑧))
140139oveqd 7448 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) = (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟))
141 oveq1 7438 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐻𝑧) = (𝑦𝐻𝑧))
142140, 141eleq12d 2833 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) ∈ (𝑥𝐻𝑧) ↔ (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))
143137, 142imbi12d 344 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) ∈ (𝑥𝐻𝑧)) ↔ ((𝜑 ∧ ((𝑦𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧))))
144 df-3an 1088 . . . . . . . . . . . . . . 15 (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
14521, 144bitri 275 . . . . . . . . . . . . . 14 (𝜓 ↔ (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
146 simpll 767 . . . . . . . . . . . . . . . . . . 19 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑦 = 𝑎)
147146eleq1d 2824 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑦𝐵𝑎𝐵))
148147anbi2d 630 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ (𝑥𝐵𝑎𝐵)))
149 simplr 769 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑤 = 𝑧)
150149eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑤𝐵𝑧𝐵))
151150anbi2d 630 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ (𝑧𝐵𝑧𝐵)))
152 anidm 564 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐵𝑧𝐵) ↔ 𝑧𝐵)
153151, 152bitrdi 287 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ 𝑧𝐵))
154148, 153anbi12d 632 . . . . . . . . . . . . . . . 16 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ↔ ((𝑥𝐵𝑎𝐵) ∧ 𝑧𝐵)))
155 df-3an 1088 . . . . . . . . . . . . . . . 16 ((𝑥𝐵𝑎𝐵𝑧𝐵) ↔ ((𝑥𝐵𝑎𝐵) ∧ 𝑧𝐵))
156154, 155bitr4di 289 . . . . . . . . . . . . . . 15 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ↔ (𝑥𝐵𝑎𝐵𝑧𝐵)))
157 simpr 484 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑓 = 𝑟)
158146oveq2d 7447 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑥𝐻𝑦) = (𝑥𝐻𝑎))
159157, 158eleq12d 2833 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑟 ∈ (𝑥𝐻𝑎)))
160146oveq1d 7446 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑦𝐻𝑧) = (𝑎𝐻𝑧))
161160eleq2d 2825 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑔 ∈ (𝑎𝐻𝑧)))
162149oveq2d 7447 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑧𝐻𝑤) = (𝑧𝐻𝑧))
163162eleq2d 2825 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑘 ∈ (𝑧𝐻𝑤) ↔ 𝑘 ∈ (𝑧𝐻𝑧)))
164159, 161, 1633anbi123d 1435 . . . . . . . . . . . . . . . 16 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))
165 df-3an 1088 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑧)) ↔ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))
166164, 165bitrdi 287 . . . . . . . . . . . . . . 15 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))
167156, 166anbi12d 632 . . . . . . . . . . . . . 14 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))))
168145, 167bitrid 283 . . . . . . . . . . . . 13 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝜓 ↔ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))))
169168anbi2d 630 . . . . . . . . . . . 12 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))))
170146opeq2d 4885 . . . . . . . . . . . . . . 15 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑎⟩)
171170oveq1d 7446 . . . . . . . . . . . . . 14 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (⟨𝑥, 𝑦· 𝑧) = (⟨𝑥, 𝑎· 𝑧))
172 eqidd 2736 . . . . . . . . . . . . . 14 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑔 = 𝑔)
173171, 172, 157oveq123d 7452 . . . . . . . . . . . . 13 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟))
174173eleq1d 2824 . . . . . . . . . . . 12 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ↔ (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) ∈ (𝑥𝐻𝑧)))
175169, 174imbi12d 344 . . . . . . . . . . 11 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ↔ ((𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) ∈ (𝑥𝐻𝑧))))
176175sbiedvw 2093 . . . . . . . . . 10 ((𝑦 = 𝑎𝑤 = 𝑧) → ([𝑟 / 𝑓]((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ↔ ((𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) ∈ (𝑥𝐻𝑧))))
177176sbiedvw 2093 . . . . . . . . 9 (𝑦 = 𝑎 → ([𝑧 / 𝑤][𝑟 / 𝑓]((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ↔ ((𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) ∈ (𝑥𝐻𝑧))))
178 iscatd2.4 . . . . . . . . . . 11 ((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
179178sbt 2064 . . . . . . . . . 10 [𝑟 / 𝑓]((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
180179sbt 2064 . . . . . . . . 9 [𝑧 / 𝑤][𝑟 / 𝑓]((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
181177, 180chvarvv 1996 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) ∈ (𝑥𝐻𝑧))
182143, 181chvarvv 1996 . . . . . . 7 ((𝜑 ∧ ((𝑦𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧))
183182exp45 438 . . . . . 6 (𝜑 → ((𝑦𝐵𝑎𝐵𝑧𝐵) → ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) → (𝑘 ∈ (𝑧𝐻𝑧) → (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))))
1841833imp 1110 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (𝑘 ∈ (𝑧𝐻𝑧) → (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))
185184exlimdv 1931 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (∃𝑘 𝑘 ∈ (𝑧𝐻𝑧) → (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))
186129, 185mpd 15 . . 3 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧))
187130anbi1d 631 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝐵𝑎𝐵) ↔ (𝑦𝐵𝑎𝐵)))
188187anbi1d 631 . . . . . 6 (𝑥 = 𝑦 → (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ↔ ((𝑦𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵))))
1891333anbi1d 1439 . . . . . 6 (𝑥 = 𝑦 → ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
190188, 1893anbi23d 1438 . . . . 5 (𝑥 = 𝑦 → ((𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (𝜑 ∧ ((𝑦𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
191138oveq1d 7446 . . . . . . 7 (𝑥 = 𝑦 → (⟨𝑥, 𝑎· 𝑤) = (⟨𝑦, 𝑎· 𝑤))
192191oveqd 7448 . . . . . 6 (𝑥 = 𝑦 → ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟) = ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑦, 𝑎· 𝑤)𝑟))
193 opeq1 4878 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
194193oveq1d 7446 . . . . . . 7 (𝑥 = 𝑦 → (⟨𝑥, 𝑧· 𝑤) = (⟨𝑦, 𝑧· 𝑤))
195 eqidd 2736 . . . . . . 7 (𝑥 = 𝑦𝑘 = 𝑘)
196194, 195, 140oveq123d 7452 . . . . . 6 (𝑥 = 𝑦 → (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟)) = (𝑘(⟨𝑦, 𝑧· 𝑤)(𝑔(⟨𝑦, 𝑎· 𝑧)𝑟)))
197192, 196eqeq12d 2751 . . . . 5 (𝑥 = 𝑦 → (((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟)) ↔ ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑦, 𝑎· 𝑤)𝑟) = (𝑘(⟨𝑦, 𝑧· 𝑤)(𝑔(⟨𝑦, 𝑎· 𝑧)𝑟))))
198190, 197imbi12d 344 . . . 4 (𝑥 = 𝑦 → (((𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟))) ↔ ((𝜑 ∧ ((𝑦𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑦, 𝑎· 𝑤)𝑟) = (𝑘(⟨𝑦, 𝑧· 𝑤)(𝑔(⟨𝑦, 𝑎· 𝑧)𝑟)))))
199 simpl 482 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑓 = 𝑟) → 𝑦 = 𝑎)
200199eleq1d 2824 . . . . . . . . . . . . 13 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑦𝐵𝑎𝐵))
201200anbi2d 630 . . . . . . . . . . . 12 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ (𝑥𝐵𝑎𝐵)))
202 simpr 484 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑓 = 𝑟) → 𝑓 = 𝑟)
203199oveq2d 7447 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑥𝐻𝑦) = (𝑥𝐻𝑎))
204202, 203eleq12d 2833 . . . . . . . . . . . . 13 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑟 ∈ (𝑥𝐻𝑎)))
205199oveq1d 7446 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑦𝐻𝑧) = (𝑎𝐻𝑧))
206205eleq2d 2825 . . . . . . . . . . . . 13 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑔 ∈ (𝑎𝐻𝑧)))
207204, 2063anbi12d 1436 . . . . . . . . . . . 12 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
208201, 2073anbi13d 1437 . . . . . . . . . . 11 ((𝑦 = 𝑎𝑓 = 𝑟) → (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
20921, 208bitrid 283 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝜓 ↔ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
210 df-3an 1088 . . . . . . . . . 10 (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
211209, 210bitrdi 287 . . . . . . . . 9 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝜓 ↔ (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
212211anbi2d 630 . . . . . . . 8 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))))
213 3anass 1094 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (𝜑 ∧ (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
214212, 213bitr4di 289 . . . . . . 7 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
215199opeq2d 4885 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑎⟩)
216215oveq1d 7446 . . . . . . . . 9 ((𝑦 = 𝑎𝑓 = 𝑟) → (⟨𝑥, 𝑦· 𝑤) = (⟨𝑥, 𝑎· 𝑤))
217199opeq1d 4884 . . . . . . . . . . 11 ((𝑦 = 𝑎𝑓 = 𝑟) → ⟨𝑦, 𝑧⟩ = ⟨𝑎, 𝑧⟩)
218217oveq1d 7446 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → (⟨𝑦, 𝑧· 𝑤) = (⟨𝑎, 𝑧· 𝑤))
219218oveqd 7448 . . . . . . . . 9 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑘(⟨𝑦, 𝑧· 𝑤)𝑔) = (𝑘(⟨𝑎, 𝑧· 𝑤)𝑔))
220216, 219, 202oveq123d 7452 . . . . . . . 8 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟))
221215oveq1d 7446 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → (⟨𝑥, 𝑦· 𝑧) = (⟨𝑥, 𝑎· 𝑧))
222 eqidd 2736 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → 𝑔 = 𝑔)
223221, 222, 202oveq123d 7452 . . . . . . . . 9 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟))
224223oveq2d 7447 . . . . . . . 8 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟)))
225220, 224eqeq12d 2751 . . . . . . 7 ((𝑦 = 𝑎𝑓 = 𝑟) → (((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)) ↔ ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟))))
226214, 225imbi12d 344 . . . . . 6 ((𝑦 = 𝑎𝑓 = 𝑟) → (((𝜑𝜓) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) ↔ ((𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟)))))
227226sbiedvw 2093 . . . . 5 (𝑦 = 𝑎 → ([𝑟 / 𝑓]((𝜑𝜓) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) ↔ ((𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟)))))
228 iscatd2.5 . . . . . 6 ((𝜑𝜓) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
229228sbt 2064 . . . . 5 [𝑟 / 𝑓]((𝜑𝜓) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
230227, 229chvarvv 1996 . . . 4 ((𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟)))
231198, 230chvarvv 1996 . . 3 ((𝜑 ∧ ((𝑦𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑦, 𝑎· 𝑤)𝑟) = (𝑘(⟨𝑦, 𝑧· 𝑤)(𝑔(⟨𝑦, 𝑎· 𝑧)𝑟)))
2321, 2, 3, 4, 5, 61, 121, 186, 231iscatd 17718 . 2 (𝜑𝐶 ∈ Cat)
2331, 2, 3, 232, 5, 61, 121catidd 17725 . 2 (𝜑 → (Id‘𝐶) = (𝑦𝐵1 ))
234232, 233jca 511 1 (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦𝐵1 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  [wsb 2062  wcel 2106  wne 2938  wral 3059  c0 4339  cop 4637  cmpt 5231  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-cat 17713  df-cid 17714
This theorem is referenced by:  oppccatid  17766  subccatid  17897  fuccatid  18026  setccatid  18138  catccatid  18160  estrccatid  18187  xpccatid  18244  rngccatidALTV  48116  ringccatidALTV  48150  isthincd2  48838  mndtccatid  48896
  Copyright terms: Public domain W3C validator