Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funcringcsetclem8ALTV Structured version   Visualization version   GIF version
Theorem funcringcsetclem8ALTV 47494
Description: Lemma 8 for funcringcsetcALTV 47496. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV.r 𝑅 = (RingCatALTVβ€˜π‘ˆ)
funcringcsetcALTV.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcringcsetcALTV.b 𝐡 = (Baseβ€˜π‘…)
funcringcsetcALTV.c 𝐢 = (Baseβ€˜π‘†)
funcringcsetcALTV.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcringcsetcALTV.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcringcsetcALTV.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetclem8ALTV ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ):(𝑋(Hom β€˜π‘…)π‘Œ)⟢((πΉβ€˜π‘‹)(Hom β€˜π‘†)(πΉβ€˜π‘Œ)))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝑋   πœ‘,π‘₯   π‘₯,𝐢   𝑦,𝐡,π‘₯   𝑦,𝑋   π‘₯,π‘Œ,𝑦   πœ‘,𝑦
Allowed substitution hints:   𝐢(𝑦)   𝑅(π‘₯,𝑦)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)
Proof of Theorem funcringcsetclem8ALTV
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1oi 6874 . . . 4 ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)–1-1-ontoβ†’(𝑋 RingHom π‘Œ)
2 f1of 6836 . . . 4 (( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)–1-1-ontoβ†’(𝑋 RingHom π‘Œ) β†’ ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)⟢(𝑋 RingHom π‘Œ))
31, 2mp1i 13 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)⟢(𝑋 RingHom π‘Œ))
4 eqid 2725 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
5 eqid 2725 . . . . . 6 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
64, 5rhmf 20428 . . . . 5 (𝑓 ∈ (𝑋 RingHom π‘Œ) β†’ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
7 fvex 6907 . . . . . . . . . 10 (Baseβ€˜π‘Œ) ∈ V
8 fvex 6907 . . . . . . . . . 10 (Baseβ€˜π‘‹) ∈ V
97, 8pm3.2i 469 . . . . . . . . 9 ((Baseβ€˜π‘Œ) ∈ V ∧ (Baseβ€˜π‘‹) ∈ V)
10 elmapg 8856 . . . . . . . . . 10 (((Baseβ€˜π‘Œ) ∈ V ∧ (Baseβ€˜π‘‹) ∈ V) β†’ (𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↔ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)))
1110bicomd 222 . . . . . . . . 9 (((Baseβ€˜π‘Œ) ∈ V ∧ (Baseβ€˜π‘‹) ∈ V) β†’ (𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ↔ 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))))
129, 11mp1i 13 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ↔ 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))))
1312biimpa 475 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)) β†’ 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
14 simpr 483 . . . . . . . . . 10 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
15 funcringcsetcALTV.r . . . . . . . . . . 11 𝑅 = (RingCatALTVβ€˜π‘ˆ)
16 funcringcsetcALTV.s . . . . . . . . . . 11 𝑆 = (SetCatβ€˜π‘ˆ)
17 funcringcsetcALTV.b . . . . . . . . . . 11 𝐡 = (Baseβ€˜π‘…)
18 funcringcsetcALTV.c . . . . . . . . . . 11 𝐢 = (Baseβ€˜π‘†)
19 funcringcsetcALTV.u . . . . . . . . . . 11 (πœ‘ β†’ π‘ˆ ∈ WUni)
20 funcringcsetcALTV.f . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
2115, 16, 17, 18, 19, 20funcringcsetclem1ALTV 47487 . . . . . . . . . 10 ((πœ‘ ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
2214, 21sylan2 591 . . . . . . . . 9 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (πΉβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
23 simpl 481 . . . . . . . . . 10 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
2415, 16, 17, 18, 19, 20funcringcsetclem1ALTV 47487 . . . . . . . . . 10 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
2523, 24sylan2 591 . . . . . . . . 9 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
2622, 25oveq12d 7435 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)) = ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
2726adantr 479 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)) β†’ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)) = ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
2813, 27eleqtrrd 2828 . . . . . 6 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)) β†’ 𝑓 ∈ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)))
2928ex 411 . . . . 5 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) β†’ 𝑓 ∈ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹))))
306, 29syl5 34 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑓 ∈ (𝑋 RingHom π‘Œ) β†’ 𝑓 ∈ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹))))
3130ssrdv 3983 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋 RingHom π‘Œ) βŠ† ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)))
323, 31fssd 6738 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)⟢((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)))
33 funcringcsetcALTV.g . . . 4 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
3415, 16, 17, 18, 19, 20, 33funcringcsetclem5ALTV 47491 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ) = ( I β†Ύ (𝑋 RingHom π‘Œ)))
3519adantr 479 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
36 eqid 2725 . . . 4 (Hom β€˜π‘…) = (Hom β€˜π‘…)
3723adantl 480 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ 𝑋 ∈ 𝐡)
3814adantl 480 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ π‘Œ ∈ 𝐡)
3915, 17, 35, 36, 37, 38ringchomALTV 47476 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋(Hom β€˜π‘…)π‘Œ) = (𝑋 RingHom π‘Œ))
40 eqid 2725 . . . 4 (Hom β€˜π‘†) = (Hom β€˜π‘†)
4115, 16, 17, 18, 19, 20funcringcsetclem2ALTV 47488 . . . . 5 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) ∈ π‘ˆ)
4223, 41sylan2 591 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (πΉβ€˜π‘‹) ∈ π‘ˆ)
4315, 16, 17, 18, 19, 20funcringcsetclem2ALTV 47488 . . . . 5 ((πœ‘ ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜π‘Œ) ∈ π‘ˆ)
4414, 43sylan2 591 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (πΉβ€˜π‘Œ) ∈ π‘ˆ)
4516, 35, 40, 42, 44setchom 18068 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((πΉβ€˜π‘‹)(Hom β€˜π‘†)(πΉβ€˜π‘Œ)) = ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)))
4634, 39, 45feq123d 6710 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((π‘‹πΊπ‘Œ):(𝑋(Hom β€˜π‘…)π‘Œ)⟢((πΉβ€˜π‘‹)(Hom β€˜π‘†)(πΉβ€˜π‘Œ)) ↔ ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)⟢((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹))))
4732, 46mpbird 256 1 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ):(𝑋(Hom β€˜π‘…)π‘Œ)⟢((πΉβ€˜π‘‹)(Hom β€˜π‘†)(πΉβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ↦ cmpt 5231   I cid 5574   β†Ύ cres 5679  βŸΆwf 6543  β€“1-1-ontoβ†’wf1o 6546  β€˜cfv 6547  (class class class)co 7417   ∈ cmpo 7419   ↑m cmap 8843  WUnicwun 10723  Basecbs 17179  Hom chom 17243  SetCatcsetc 18063   RingHom crh 20412  RingCatALTVcringcALTV 47461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  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-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7373  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-wun 10725  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-plusg 17245  df-hom 17256  df-cco 17257  df-0g 17422  df-setc 18064  df-mhm 18739  df-ghm 19172  df-mgp 20079  df-ur 20126  df-ring 20179  df-rhm 20415  df-ringcALTV 47462
This theorem is referenced by:  funcringcsetcALTV  47496
  Copyright terms: Public domain W3C validator