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Theorem initoeu2lem0 17963
Description: Lemma 0 for initoeu2 17966. (Contributed by AV, 9-Apr-2020.)
Hypotheses
Ref Expression
initoeu1.c (πœ‘ β†’ 𝐢 ∈ Cat)
initoeu1.a (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))
initoeu2lem.x 𝑋 = (Baseβ€˜πΆ)
initoeu2lem.h 𝐻 = (Hom β€˜πΆ)
initoeu2lem.i 𝐼 = (Isoβ€˜πΆ)
initoeu2lem.o ⚬ = (compβ€˜πΆ)
Assertion
Ref Expression
initoeu2lem0 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) ∧ ((𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) = (𝐺(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ))) β†’ 𝐺 = (𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾))

Proof of Theorem initoeu2lem0
StepHypRef Expression
1 3simpa 1149 . 2 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) ∧ ((𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) = (𝐺(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ))) β†’ ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))))
2 simp3 1139 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) ∧ ((𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) = (𝐺(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ))) β†’ ((𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) = (𝐺(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)))
32eqcomd 2739 . 2 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) ∧ ((𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) = (𝐺(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ))) β†’ (𝐺(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) = ((𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)))
4 initoeu2lem.x . . 3 𝑋 = (Baseβ€˜πΆ)
5 eqid 2733 . . 3 (Invβ€˜πΆ) = (Invβ€˜πΆ)
6 initoeu1.c . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
76adantr 482 . . . 4 ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ 𝐢 ∈ Cat)
87adantr 482 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐢 ∈ Cat)
9 simpr1 1195 . . . 4 ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
109adantr 482 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐴 ∈ 𝑋)
11 simpr2 1196 . . . 4 ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
1211adantr 482 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐡 ∈ 𝑋)
13 simplr3 1218 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐷 ∈ 𝑋)
14 initoeu2lem.i . . . . . . . 8 𝐼 = (Isoβ€˜πΆ)
1514oveqi 7422 . . . . . . 7 (𝐡𝐼𝐴) = (𝐡(Isoβ€˜πΆ)𝐴)
1615eleq2i 2826 . . . . . 6 (𝐾 ∈ (𝐡𝐼𝐴) ↔ 𝐾 ∈ (𝐡(Isoβ€˜πΆ)𝐴))
1716biimpi 215 . . . . 5 (𝐾 ∈ (𝐡𝐼𝐴) β†’ 𝐾 ∈ (𝐡(Isoβ€˜πΆ)𝐴))
18173ad2ant1 1134 . . . 4 ((𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) β†’ 𝐾 ∈ (𝐡(Isoβ€˜πΆ)𝐴))
1918adantl 483 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐾 ∈ (𝐡(Isoβ€˜πΆ)𝐴))
20 initoeu2lem.h . . . . . . . 8 𝐻 = (Hom β€˜πΆ)
2120oveqi 7422 . . . . . . 7 (𝐡𝐻𝐷) = (𝐡(Hom β€˜πΆ)𝐷)
2221eleq2i 2826 . . . . . 6 (𝐺 ∈ (𝐡𝐻𝐷) ↔ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐷))
2322biimpi 215 . . . . 5 (𝐺 ∈ (𝐡𝐻𝐷) β†’ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐷))
24233ad2ant3 1136 . . . 4 ((𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) β†’ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐷))
2524adantl 483 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐷))
26 eqid 2733 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
27 initoeu2lem.o . . . 4 ⚬ = (compβ€˜πΆ)
284, 26, 14, 7, 11, 9isohom 17723 . . . . . . . 8 ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ (𝐡𝐼𝐴) βŠ† (𝐡(Hom β€˜πΆ)𝐴))
2928sseld 3982 . . . . . . 7 ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ (𝐾 ∈ (𝐡𝐼𝐴) β†’ 𝐾 ∈ (𝐡(Hom β€˜πΆ)𝐴)))
3029com12 32 . . . . . 6 (𝐾 ∈ (𝐡𝐼𝐴) β†’ ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ 𝐾 ∈ (𝐡(Hom β€˜πΆ)𝐴)))
31303ad2ant1 1134 . . . . 5 ((𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) β†’ ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ 𝐾 ∈ (𝐡(Hom β€˜πΆ)𝐴)))
3231impcom 409 . . . 4 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐾 ∈ (𝐡(Hom β€˜πΆ)𝐴))
3320oveqi 7422 . . . . . . . 8 (𝐴𝐻𝐷) = (𝐴(Hom β€˜πΆ)𝐷)
3433eleq2i 2826 . . . . . . 7 (𝐹 ∈ (𝐴𝐻𝐷) ↔ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐷))
3534biimpi 215 . . . . . 6 (𝐹 ∈ (𝐴𝐻𝐷) β†’ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐷))
36353ad2ant2 1135 . . . . 5 ((𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) β†’ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐷))
3736adantl 483 . . . 4 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐷))
384, 26, 27, 8, 12, 10, 13, 32, 37catcocl 17629 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ (𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐡(Hom β€˜πΆ)𝐷))
39 eqid 2733 . . 3 ((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ) = ((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)
4027oveqi 7422 . . 3 (⟨𝐴, 𝐡⟩ ⚬ 𝐷) = (⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐷)
414, 5, 8, 10, 12, 13, 19, 25, 38, 39, 40rcaninv 17741 . 2 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ ((𝐺(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) = ((𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) β†’ 𝐺 = (𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)))
421, 3, 41sylc 65 1 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) ∧ ((𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) = (𝐺(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ))) β†’ 𝐺 = (𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4635  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Hom chom 17208  compcco 17209  Catccat 17608  Invcinv 17692  Isociso 17693  InitOcinito 17931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-cat 17612  df-cid 17613  df-sect 17694  df-inv 17695  df-iso 17696
This theorem is referenced by:  initoeu2lem1  17964
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