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Theorem initoeu2lem0 17967
Description: Lemma 0 for initoeu2 17970. (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 1146 . 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 1136 . . 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 2736 . 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 2730 . . 3 (Invβ€˜πΆ) = (Invβ€˜πΆ)
6 initoeu1.c . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
76adantr 479 . . . 4 ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ 𝐢 ∈ Cat)
87adantr 479 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐢 ∈ Cat)
9 simpr1 1192 . . . 4 ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ 𝐴 ∈ 𝑋)
109adantr 479 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐴 ∈ 𝑋)
11 simpr2 1193 . . . 4 ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
1211adantr 479 . . 3 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐡 ∈ 𝑋)
13 simplr3 1215 . . 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 7424 . . . . . . 7 (𝐡𝐼𝐴) = (𝐡(Isoβ€˜πΆ)𝐴)
1615eleq2i 2823 . . . . . 6 (𝐾 ∈ (𝐡𝐼𝐴) ↔ 𝐾 ∈ (𝐡(Isoβ€˜πΆ)𝐴))
1716biimpi 215 . . . . 5 (𝐾 ∈ (𝐡𝐼𝐴) β†’ 𝐾 ∈ (𝐡(Isoβ€˜πΆ)𝐴))
18173ad2ant1 1131 . . . 4 ((𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) β†’ 𝐾 ∈ (𝐡(Isoβ€˜πΆ)𝐴))
1918adantl 480 . . 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 7424 . . . . . . 7 (𝐡𝐻𝐷) = (𝐡(Hom β€˜πΆ)𝐷)
2221eleq2i 2823 . . . . . 6 (𝐺 ∈ (𝐡𝐻𝐷) ↔ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐷))
2322biimpi 215 . . . . 5 (𝐺 ∈ (𝐡𝐻𝐷) β†’ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐷))
24233ad2ant3 1133 . . . 4 ((𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) β†’ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐷))
2524adantl 480 . . 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 2730 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
27 initoeu2lem.o . . . 4 ⚬ = (compβ€˜πΆ)
284, 26, 14, 7, 11, 9isohom 17727 . . . . . . . 8 ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ (𝐡𝐼𝐴) βŠ† (𝐡(Hom β€˜πΆ)𝐴))
2928sseld 3980 . . . . . . 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 1131 . . . . 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 406 . . . 4 (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷))) β†’ 𝐾 ∈ (𝐡(Hom β€˜πΆ)𝐴))
3320oveqi 7424 . . . . . . . 8 (𝐴𝐻𝐷) = (𝐴(Hom β€˜πΆ)𝐷)
3433eleq2i 2823 . . . . . . 7 (𝐹 ∈ (𝐴𝐻𝐷) ↔ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐷))
3534biimpi 215 . . . . . 6 (𝐹 ∈ (𝐴𝐻𝐷) β†’ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐷))
36353ad2ant2 1132 . . . . 5 ((𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) β†’ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐷))
3736adantl 480 . . . 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 17633 . . 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 2730 . . 3 ((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ) = ((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)
4027oveqi 7424 . . 3 (⟨𝐴, 𝐡⟩ ⚬ 𝐷) = (⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐷)
414, 5, 8, 10, 12, 13, 19, 25, 38, 39, 40rcaninv 17745 . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βŸ¨cop 4633  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  Hom chom 17212  compcco 17213  Catccat 17612  Invcinv 17696  Isociso 17697  InitOcinito 17935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-cat 17616  df-cid 17617  df-sect 17698  df-inv 17699  df-iso 17700
This theorem is referenced by:  initoeu2lem1  17968
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