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Theorem iscatd2 17561
Description: Version of iscatd 17553 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 4295 . . . . . 6 ((𝜑𝑦𝐵) → (𝑦𝐻𝑦) ≠ ∅)
763ad2antr1 1188 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → (𝑦𝐻𝑦) ≠ ∅)
8 n0 4306 . . . . 5 ((𝑦𝐻𝑦) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑦𝐻𝑦))
97, 8sylib 217 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ∃𝑔 𝑔 ∈ (𝑦𝐻𝑦))
10 n0 4306 . . . . 5 ((𝑦𝐻𝑦) ≠ ∅ ↔ ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦))
117, 10sylib 217 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦))
12 exdistrv 1959 . . . . 5 (∃𝑔𝑘(𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)) ↔ (∃𝑔 𝑔 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦)))
13 simpll 765 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝜑)
14 simplr2 1216 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑎𝐵)
15 simplr1 1215 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑦𝐵)
1614, 15jca 512 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → (𝑎𝐵𝑦𝐵))
17 simplr3 1217 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑟 ∈ (𝑎𝐻𝑦))
18 simprl 769 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑔 ∈ (𝑦𝐻𝑦))
19 simprr 771 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → 𝑘 ∈ (𝑦𝐻𝑦))
2017, 18, 193jca 1128 . . . . . . . 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 773 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑥 = 𝑎)
2322eleq1d 2822 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑥𝐵𝑎𝐵))
2423anbi1d 630 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ (𝑎𝐵𝑦𝐵)))
25 simpllr 774 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑧 = 𝑦)
2625eleq1d 2822 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑧𝐵𝑦𝐵))
27 simplr 767 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑤 = 𝑦)
2827eleq1d 2822 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑤𝐵𝑦𝐵))
2926, 28anbi12d 631 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ (𝑦𝐵𝑦𝐵)))
30 anidm 565 . . . . . . . . . . . . . . . . 17 ((𝑦𝐵𝑦𝐵) ↔ 𝑦𝐵)
3129, 30bitrdi 286 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ 𝑦𝐵))
32 simpr 485 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 𝑓 = 𝑟)
3322oveq1d 7372 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑥𝐻𝑦) = (𝑎𝐻𝑦))
3432, 33eleq12d 2832 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑟 ∈ (𝑎𝐻𝑦)))
3525oveq2d 7373 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑦𝐻𝑧) = (𝑦𝐻𝑦))
3635eleq2d 2823 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑔 ∈ (𝑦𝐻𝑦)))
3725, 27oveq12d 7375 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑧𝐻𝑤) = (𝑦𝐻𝑦))
3837eleq2d 2823 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝑘 ∈ (𝑧𝐻𝑤) ↔ 𝑘 ∈ (𝑦𝐻𝑦)))
3934, 36, 383anbi123d 1436 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))))
4024, 31, 393anbi123d 1436 . . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . . 14 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (𝜓 ↔ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))))
4241anbi2d 629 . . . . . . . . . . . . 13 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))))))
4322opeq1d 4836 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑦⟩)
4443oveq1d 7372 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → (⟨𝑥, 𝑦· 𝑦) = (⟨𝑎, 𝑦· 𝑦))
45 eqidd 2737 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → 1 = 1 )
4644, 45, 32oveq123d 7378 . . . . . . . . . . . . . 14 ((((𝑥 = 𝑎𝑧 = 𝑦) ∧ 𝑤 = 𝑦) ∧ 𝑓 = 𝑟) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟))
4746, 32eqeq12d 2752 . . . . . . . . . . . . 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 2096 . . . . . . . . . . 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 2096 . . . . . . . . . 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 2096 . . . . . . . . 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 2069 . . . . . . . . . . 11 [𝑟 / 𝑓]((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
5453sbt 2069 . . . . . . . . . 10 [𝑦 / 𝑤][𝑟 / 𝑓]((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
5554sbt 2069 . . . . . . . . 9 [𝑦 / 𝑧][𝑦 / 𝑤][𝑟 / 𝑓]((𝜑𝜓) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
5651, 55chvarvv 2002 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝐵𝑦𝐵) ∧ 𝑦𝐵 ∧ (𝑟 ∈ (𝑎𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)
5713, 16, 15, 20, 56syl13anc 1372 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) ∧ (𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)
5857ex 413 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ((𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟))
5958exlimdvv 1937 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → (∃𝑔𝑘(𝑔 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑦𝐻𝑦)) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟))
6012, 59biimtrrid 242 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ((∃𝑔 𝑔 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑦𝐻𝑦)) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟))
619, 11, 60mp2and 697 . . 3 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑎𝐻𝑦))) → ( 1 (⟨𝑎, 𝑦· 𝑦)𝑟) = 𝑟)
6263ad2antr1 1188 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → (𝑦𝐻𝑦) ≠ ∅)
63 n0 4306 . . . . 5 ((𝑦𝐻𝑦) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑦𝐻𝑦))
6462, 63sylib 217 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ∃𝑓 𝑓 ∈ (𝑦𝐻𝑦))
65 id 22 . . . . . . . 8 (𝑦 = 𝑎𝑦 = 𝑎)
6665, 65oveq12d 7375 . . . . . . 7 (𝑦 = 𝑎 → (𝑦𝐻𝑦) = (𝑎𝐻𝑎))
6766neeq1d 3003 . . . . . 6 (𝑦 = 𝑎 → ((𝑦𝐻𝑦) ≠ ∅ ↔ (𝑎𝐻𝑎) ≠ ∅))
686ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑦𝐻𝑦) ≠ ∅)
6968adantr 481 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ∀𝑦𝐵 (𝑦𝐻𝑦) ≠ ∅)
70 simpr2 1195 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → 𝑎𝐵)
7167, 69, 70rspcdva 3582 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → (𝑎𝐻𝑎) ≠ ∅)
72 n0 4306 . . . . 5 ((𝑎𝐻𝑎) ≠ ∅ ↔ ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎))
7371, 72sylib 217 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎))
74 exdistrv 1959 . . . . 5 (∃𝑓𝑘(𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎)) ↔ (∃𝑓 𝑓 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎)))
75 simpll 765 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝜑)
76 simplr1 1215 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑦𝐵)
77 simplr2 1216 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑎𝐵)
78 simprl 769 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑓 ∈ (𝑦𝐻𝑦))
79 simplr3 1217 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑟 ∈ (𝑦𝐻𝑎))
80 simprr 771 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → 𝑘 ∈ (𝑎𝐻𝑎))
8178, 79, 803jca 1128 . . . . . . . 8 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))
82 simplll 773 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑥 = 𝑦)
8382eleq1d 2822 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑥𝐵𝑦𝐵))
8483anbi1d 630 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ (𝑦𝐵𝑦𝐵)))
8584, 30bitrdi 286 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ 𝑦𝐵))
86 simpllr 774 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑧 = 𝑎)
8786eleq1d 2822 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑧𝐵𝑎𝐵))
88 simplr 767 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑤 = 𝑎)
8988eleq1d 2822 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑤𝐵𝑎𝐵))
9087, 89anbi12d 631 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ (𝑎𝐵𝑎𝐵)))
91 anidm 565 . . . . . . . . . . . . . . . . 17 ((𝑎𝐵𝑎𝐵) ↔ 𝑎𝐵)
9290, 91bitrdi 286 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ 𝑎𝐵))
9382oveq1d 7372 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑥𝐻𝑦) = (𝑦𝐻𝑦))
9493eleq2d 2823 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑦𝐻𝑦)))
95 simpr 485 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 𝑔 = 𝑟)
9686oveq2d 7373 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑦𝐻𝑧) = (𝑦𝐻𝑎))
9795, 96eleq12d 2832 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑟 ∈ (𝑦𝐻𝑎)))
9886, 88oveq12d 7375 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑧𝐻𝑤) = (𝑎𝐻𝑎))
9998eleq2d 2823 . . . . . . . . . . . . . . . . 17 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑘 ∈ (𝑧𝐻𝑤) ↔ 𝑘 ∈ (𝑎𝐻𝑎)))
10094, 97, 993anbi123d 1436 . . . . . . . . . . . . . . . 16 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎))))
10185, 92, 1003anbi123d 1436 . . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . . 14 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝜓 ↔ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))))
103102anbi2d 629 . . . . . . . . . . . . 13 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎))))))
10486oveq2d 7373 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (⟨𝑦, 𝑦· 𝑧) = (⟨𝑦, 𝑦· 𝑎))
105 eqidd 2737 . . . . . . . . . . . . . . 15 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → 1 = 1 )
106104, 95, 105oveq123d 7378 . . . . . . . . . . . . . 14 ((((𝑥 = 𝑦𝑧 = 𝑎) ∧ 𝑤 = 𝑎) ∧ 𝑔 = 𝑟) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ))
107106, 95eqeq12d 2752 . . . . . . . . . . . . 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 2096 . . . . . . . . . . 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 2096 . . . . . . . . . 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 2096 . . . . . . . . 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 2069 . . . . . . . . . . 11 [𝑟 / 𝑔]((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
114113sbt 2069 . . . . . . . . . 10 [𝑎 / 𝑤][𝑟 / 𝑔]((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
115114sbt 2069 . . . . . . . . 9 [𝑎 / 𝑧][𝑎 / 𝑤][𝑟 / 𝑔]((𝜑𝜓) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
116111, 115chvarvv 2002 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑎𝐵 ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑘 ∈ (𝑎𝐻𝑎)))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)
11775, 76, 77, 81, 116syl13anc 1372 . . . . . . 7 (((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) ∧ (𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)
118117ex 413 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ((𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎)) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟))
119118exlimdvv 1937 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → (∃𝑓𝑘(𝑓 ∈ (𝑦𝐻𝑦) ∧ 𝑘 ∈ (𝑎𝐻𝑎)) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟))
12074, 119biimtrrid 242 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → ((∃𝑓 𝑓 ∈ (𝑦𝐻𝑦) ∧ ∃𝑘 𝑘 ∈ (𝑎𝐻𝑎)) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟))
12164, 73, 120mp2and 697 . . 3 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑟 ∈ (𝑦𝐻𝑎))) → (𝑟(⟨𝑦, 𝑦· 𝑎) 1 ) = 𝑟)
122 id 22 . . . . . . . 8 (𝑦 = 𝑧𝑦 = 𝑧)
123122, 122oveq12d 7375 . . . . . . 7 (𝑦 = 𝑧 → (𝑦𝐻𝑦) = (𝑧𝐻𝑧))
124123neeq1d 3003 . . . . . 6 (𝑦 = 𝑧 → ((𝑦𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑧) ≠ ∅))
125683ad2ant1 1133 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → ∀𝑦𝐵 (𝑦𝐻𝑦) ≠ ∅)
126 simp23 1208 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → 𝑧𝐵)
127124, 125, 126rspcdva 3582 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (𝑧𝐻𝑧) ≠ ∅)
128 n0 4306 . . . . 5 ((𝑧𝐻𝑧) ≠ ∅ ↔ ∃𝑘 𝑘 ∈ (𝑧𝐻𝑧))
129127, 128sylib 217 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → ∃𝑘 𝑘 ∈ (𝑧𝐻𝑧))
130 eleq1w 2820 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
1311303anbi1d 1440 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥𝐵𝑎𝐵𝑧𝐵) ↔ (𝑦𝐵𝑎𝐵𝑧𝐵)))
132 oveq1 7364 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑥𝐻𝑎) = (𝑦𝐻𝑎))
133132eleq2d 2823 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑟 ∈ (𝑥𝐻𝑎) ↔ 𝑟 ∈ (𝑦𝐻𝑎)))
134133anbi1d 630 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ↔ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))))
135134anbi1d 630 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)) ↔ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))
136131, 135anbi12d 631 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))) ↔ ((𝑦𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))))
137136anbi2d 629 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) ↔ (𝜑 ∧ ((𝑦𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))))
138 opeq1 4830 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ⟨𝑥, 𝑎⟩ = ⟨𝑦, 𝑎⟩)
139138oveq1d 7372 . . . . . . . . . . 11 (𝑥 = 𝑦 → (⟨𝑥, 𝑎· 𝑧) = (⟨𝑦, 𝑎· 𝑧))
140139oveqd 7374 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) = (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟))
141 oveq1 7364 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐻𝑧) = (𝑦𝐻𝑧))
142140, 141eleq12d 2832 . . . . . . . . 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 1089 . . . . . . . . . . . . . . 15 (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
14521, 144bitri 274 . . . . . . . . . . . . . 14 (𝜓 ↔ (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
146 simpll 765 . . . . . . . . . . . . . . . . . . 19 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑦 = 𝑎)
147146eleq1d 2822 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑦𝐵𝑎𝐵))
148147anbi2d 629 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ (𝑥𝐵𝑎𝐵)))
149 simplr 767 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑤 = 𝑧)
150149eleq1d 2822 . . . . . . . . . . . . . . . . . . 19 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑤𝐵𝑧𝐵))
151150anbi2d 629 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ (𝑧𝐵𝑧𝐵)))
152 anidm 565 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐵𝑧𝐵) ↔ 𝑧𝐵)
153151, 152bitrdi 286 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑧𝐵𝑤𝐵) ↔ 𝑧𝐵))
154148, 153anbi12d 631 . . . . . . . . . . . . . . . 16 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ↔ ((𝑥𝐵𝑎𝐵) ∧ 𝑧𝐵)))
155 df-3an 1089 . . . . . . . . . . . . . . . 16 ((𝑥𝐵𝑎𝐵𝑧𝐵) ↔ ((𝑥𝐵𝑎𝐵) ∧ 𝑧𝐵))
156154, 155bitr4di 288 . . . . . . . . . . . . . . 15 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ↔ (𝑥𝐵𝑎𝐵𝑧𝐵)))
157 simpr 485 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑓 = 𝑟)
158146oveq2d 7373 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑥𝐻𝑦) = (𝑥𝐻𝑎))
159157, 158eleq12d 2832 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑟 ∈ (𝑥𝐻𝑎)))
160146oveq1d 7372 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑦𝐻𝑧) = (𝑎𝐻𝑧))
161160eleq2d 2823 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑔 ∈ (𝑎𝐻𝑧)))
162149oveq2d 7373 . . . . . . . . . . . . . . . . . 18 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑧𝐻𝑤) = (𝑧𝐻𝑧))
163162eleq2d 2823 . . . . . . . . . . . . . . . . 17 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑘 ∈ (𝑧𝐻𝑤) ↔ 𝑘 ∈ (𝑧𝐻𝑧)))
164159, 161, 1633anbi123d 1436 . . . . . . . . . . . . . . . 16 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))
165 df-3an 1089 . . . . . . . . . . . . . . . 16 ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑧)) ↔ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))
166164, 165bitrdi 286 . . . . . . . . . . . . . . 15 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))
167156, 166anbi12d 631 . . . . . . . . . . . . . 14 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))))
168145, 167bitrid 282 . . . . . . . . . . . . 13 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝜓 ↔ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))))
169168anbi2d 629 . . . . . . . . . . . 12 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧))))))
170146opeq2d 4837 . . . . . . . . . . . . . . 15 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑎⟩)
171170oveq1d 7372 . . . . . . . . . . . . . 14 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (⟨𝑥, 𝑦· 𝑧) = (⟨𝑥, 𝑎· 𝑧))
172 eqidd 2737 . . . . . . . . . . . . . 14 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → 𝑔 = 𝑔)
173171, 172, 157oveq123d 7378 . . . . . . . . . . . . 13 (((𝑦 = 𝑎𝑤 = 𝑧) ∧ 𝑓 = 𝑟) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟))
174173eleq1d 2822 . . . . . . . . . . . 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 2096 . . . . . . . . . 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 2096 . . . . . . . . 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 2069 . . . . . . . . . 10 [𝑟 / 𝑓]((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
180179sbt 2069 . . . . . . . . 9 [𝑧 / 𝑤][𝑟 / 𝑓]((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
181177, 180chvarvv 2002 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟) ∈ (𝑥𝐻𝑧))
182143, 181chvarvv 2002 . . . . . . 7 ((𝜑 ∧ ((𝑦𝐵𝑎𝐵𝑧𝐵) ∧ ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑧)))) → (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧))
183182exp45 439 . . . . . 6 (𝜑 → ((𝑦𝐵𝑎𝐵𝑧𝐵) → ((𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧)) → (𝑘 ∈ (𝑧𝐻𝑧) → (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))))
1841833imp 1111 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑎𝐵𝑧𝐵) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧))) → (𝑘 ∈ (𝑧𝐻𝑧) → (𝑔(⟨𝑦, 𝑎· 𝑧)𝑟) ∈ (𝑦𝐻𝑧)))
185184exlimdv 1936 . . . 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 630 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝐵𝑎𝐵) ↔ (𝑦𝐵𝑎𝐵)))
188187anbi1d 630 . . . . . 6 (𝑥 = 𝑦 → (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ↔ ((𝑦𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵))))
1891333anbi1d 1440 . . . . . 6 (𝑥 = 𝑦 → ((𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
190188, 1893anbi23d 1439 . . . . 5 (𝑥 = 𝑦 → ((𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (𝜑 ∧ ((𝑦𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑦𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
191138oveq1d 7372 . . . . . . 7 (𝑥 = 𝑦 → (⟨𝑥, 𝑎· 𝑤) = (⟨𝑦, 𝑎· 𝑤))
192191oveqd 7374 . . . . . 6 (𝑥 = 𝑦 → ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟) = ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑦, 𝑎· 𝑤)𝑟))
193 opeq1 4830 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
194193oveq1d 7372 . . . . . . 7 (𝑥 = 𝑦 → (⟨𝑥, 𝑧· 𝑤) = (⟨𝑦, 𝑧· 𝑤))
195 eqidd 2737 . . . . . . 7 (𝑥 = 𝑦𝑘 = 𝑘)
196194, 195, 140oveq123d 7378 . . . . . 6 (𝑥 = 𝑦 → (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟)) = (𝑘(⟨𝑦, 𝑧· 𝑤)(𝑔(⟨𝑦, 𝑎· 𝑧)𝑟)))
197192, 196eqeq12d 2752 . . . . 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 483 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑓 = 𝑟) → 𝑦 = 𝑎)
200199eleq1d 2822 . . . . . . . . . . . . 13 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑦𝐵𝑎𝐵))
201200anbi2d 629 . . . . . . . . . . . 12 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝑥𝐵𝑦𝐵) ↔ (𝑥𝐵𝑎𝐵)))
202 simpr 485 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑓 = 𝑟) → 𝑓 = 𝑟)
203199oveq2d 7373 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑥𝐻𝑦) = (𝑥𝐻𝑎))
204202, 203eleq12d 2832 . . . . . . . . . . . . 13 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑟 ∈ (𝑥𝐻𝑎)))
205199oveq1d 7372 . . . . . . . . . . . . . 14 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑦𝐻𝑧) = (𝑎𝐻𝑧))
206205eleq2d 2823 . . . . . . . . . . . . 13 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑔 ∈ (𝑦𝐻𝑧) ↔ 𝑔 ∈ (𝑎𝐻𝑧)))
207204, 2063anbi12d 1437 . . . . . . . . . . . 12 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
208201, 2073anbi13d 1438 . . . . . . . . . . 11 ((𝑦 = 𝑎𝑓 = 𝑟) → (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
20921, 208bitrid 282 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝜓 ↔ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
210 df-3an 1089 . . . . . . . . . 10 (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
211209, 210bitrdi 286 . . . . . . . . 9 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝜓 ↔ (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
212211anbi2d 629 . . . . . . . 8 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))))
213 3anass 1095 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (𝜑 ∧ (((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
214212, 213bitr4di 288 . . . . . . 7 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝜑𝜓) ↔ (𝜑 ∧ ((𝑥𝐵𝑎𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑟 ∈ (𝑥𝐻𝑎) ∧ 𝑔 ∈ (𝑎𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))))
215199opeq2d 4837 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑎⟩)
216215oveq1d 7372 . . . . . . . . 9 ((𝑦 = 𝑎𝑓 = 𝑟) → (⟨𝑥, 𝑦· 𝑤) = (⟨𝑥, 𝑎· 𝑤))
217199opeq1d 4836 . . . . . . . . . . 11 ((𝑦 = 𝑎𝑓 = 𝑟) → ⟨𝑦, 𝑧⟩ = ⟨𝑎, 𝑧⟩)
218217oveq1d 7372 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → (⟨𝑦, 𝑧· 𝑤) = (⟨𝑎, 𝑧· 𝑤))
219218oveqd 7374 . . . . . . . . 9 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑘(⟨𝑦, 𝑧· 𝑤)𝑔) = (𝑘(⟨𝑎, 𝑧· 𝑤)𝑔))
220216, 219, 202oveq123d 7378 . . . . . . . 8 ((𝑦 = 𝑎𝑓 = 𝑟) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = ((𝑘(⟨𝑎, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑎· 𝑤)𝑟))
221215oveq1d 7372 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → (⟨𝑥, 𝑦· 𝑧) = (⟨𝑥, 𝑎· 𝑧))
222 eqidd 2737 . . . . . . . . . 10 ((𝑦 = 𝑎𝑓 = 𝑟) → 𝑔 = 𝑔)
223221, 222, 202oveq123d 7378 . . . . . . . . 9 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑎· 𝑧)𝑟))
224223oveq2d 7373 . . . . . . . 8 ((𝑦 = 𝑎𝑓 = 𝑟) → (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑎· 𝑧)𝑟)))
225220, 224eqeq12d 2752 . . . . . . 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 2096 . . . . 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 2069 . . . . 5 [𝑟 / 𝑓]((𝜑𝜓) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
230227, 229chvarvv 2002 . . . 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 2002 . . 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 17553 . 2 (𝜑𝐶 ∈ Cat)
2331, 2, 3, 232, 5, 61, 121catidd 17560 . 2 (𝜑 → (Id‘𝐶) = (𝑦𝐵1 ))
234232, 233jca 512 1 (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦𝐵1 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  [wsb 2067  wcel 2106  wne 2943  wral 3064  c0 4282  cop 4592  cmpt 5188  cfv 6496  (class class class)co 7357  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Idccid 17545
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-cat 17548  df-cid 17549
This theorem is referenced by:  oppccatid  17601  subccatid  17732  fuccatid  17858  setccatid  17970  catccatid  17992  estrccatid  18019  xpccatid  18076  rngccatidALTV  46277  ringccatidALTV  46340  isthincd2  47048  mndtccatid  47103
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