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Theorem rhmsubcOLDlem3 33059
Description: Lemma 3 for rhmsubcOLD 33061. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
rngcrescrhmOLD.u  |-  ( ph  ->  U  e.  V )
rngcrescrhmOLD.c  |-  C  =  (RngCatOLD `  U )
rngcrescrhmOLD.r  |-  ( ph  ->  R  =  ( Ring 
i^i  U ) )
rngcrescrhmOLD.h  |-  H  =  ( RingHom  |`  ( R  X.  R ) )
Assertion
Ref Expression
rhmsubcOLDlem3  |-  ( (
ph  /\  x  e.  R )  ->  (
( Id `  (RngCatOLD `  U ) ) `  x )  e.  ( x H x ) )
Distinct variable group:    x, R
Allowed substitution hints:    ph( x)    C( x)    U( x)    H( x)    V( x)

Proof of Theorem rhmsubcOLDlem3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rngcrescrhmOLD.r . . . . . 6  |-  ( ph  ->  R  =  ( Ring 
i^i  U ) )
21eleq2d 2527 . . . . 5  |-  ( ph  ->  ( x  e.  R  <->  x  e.  ( Ring  i^i  U ) ) )
3 elinel1 31627 . . . . 5  |-  ( x  e.  ( Ring  i^i  U )  ->  x  e.  Ring )
42, 3syl6bi 228 . . . 4  |-  ( ph  ->  ( x  e.  R  ->  x  e.  Ring )
)
54imp 429 . . 3  |-  ( (
ph  /\  x  e.  R )  ->  x  e.  Ring )
6 eqid 2457 . . . 4  |-  ( Base `  x )  =  (
Base `  x )
76idrhm 17507 . . 3  |-  ( x  e.  Ring  ->  (  _I  |`  ( Base `  x
) )  e.  ( x RingHom  x ) )
85, 7syl 16 . 2  |-  ( (
ph  /\  x  e.  R )  ->  (  _I  |`  ( Base `  x
) )  e.  ( x RingHom  x ) )
9 rngcrescrhmOLD.u . . . . 5  |-  ( ph  ->  U  e.  V )
109adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  R )  ->  U  e.  V )
11 eqid 2457 . . . . 5  |-  (RngCatOLD `  U
)  =  (RngCatOLD `  U
)
12 eqid 2457 . . . . 5  |-  ( Base `  (RngCatOLD `  U )
)  =  ( Base `  (RngCatOLD `  U )
)
1311, 12rngccatidOLD 32941 . . . 4  |-  ( U  e.  V  ->  (
(RngCatOLD `  U )  e. 
Cat  /\  ( Id `  (RngCatOLD `  U )
)  =  ( y  e.  ( Base `  (RngCatOLD `  U ) )  |->  (  _I  |`  ( Base `  y ) ) ) ) )
14 simpr 461 . . . 4  |-  ( ( (RngCatOLD `  U )  e.  Cat  /\  ( Id
`  (RngCatOLD `  U )
)  =  ( y  e.  ( Base `  (RngCatOLD `  U ) )  |->  (  _I  |`  ( Base `  y ) ) ) )  ->  ( Id `  (RngCatOLD `  U )
)  =  ( y  e.  ( Base `  (RngCatOLD `  U ) )  |->  (  _I  |`  ( Base `  y ) ) ) )
1510, 13, 143syl 20 . . 3  |-  ( (
ph  /\  x  e.  R )  ->  ( Id `  (RngCatOLD `  U
) )  =  ( y  e.  ( Base `  (RngCatOLD `  U )
)  |->  (  _I  |`  ( Base `  y ) ) ) )
16 fveq2 5872 . . . . 5  |-  ( y  =  x  ->  ( Base `  y )  =  ( Base `  x
) )
1716reseq2d 5283 . . . 4  |-  ( y  =  x  ->  (  _I  |`  ( Base `  y
) )  =  (  _I  |`  ( Base `  x ) ) )
1817adantl 466 . . 3  |-  ( ( ( ph  /\  x  e.  R )  /\  y  =  x )  ->  (  _I  |`  ( Base `  y
) )  =  (  _I  |`  ( Base `  x ) ) )
19 incom 3687 . . . . . . . 8  |-  ( Ring 
i^i  U )  =  ( U  i^i  Ring )
201, 19syl6eq 2514 . . . . . . 7  |-  ( ph  ->  R  =  ( U  i^i  Ring ) )
2120eleq2d 2527 . . . . . 6  |-  ( ph  ->  ( x  e.  R  <->  x  e.  ( U  i^i  Ring ) ) )
22 ringrng 32829 . . . . . . . 8  |-  ( x  e.  Ring  ->  x  e. Rng )
2322anim2i 569 . . . . . . 7  |-  ( ( x  e.  U  /\  x  e.  Ring )  -> 
( x  e.  U  /\  x  e. Rng )
)
24 elin 3683 . . . . . . 7  |-  ( x  e.  ( U  i^i  Ring )  <->  ( x  e.  U  /\  x  e. 
Ring ) )
25 elin 3683 . . . . . . 7  |-  ( x  e.  ( U  i^i Rng )  <-> 
( x  e.  U  /\  x  e. Rng )
)
2623, 24, 253imtr4i 266 . . . . . 6  |-  ( x  e.  ( U  i^i  Ring )  ->  x  e.  ( U  i^i Rng ) )
2721, 26syl6bi 228 . . . . 5  |-  ( ph  ->  ( x  e.  R  ->  x  e.  ( U  i^i Rng ) ) )
2827imp 429 . . . 4  |-  ( (
ph  /\  x  e.  R )  ->  x  e.  ( U  i^i Rng )
)
29 rngcrescrhmOLD.c . . . . . 6  |-  C  =  (RngCatOLD `  U )
3029eqcomi 2470 . . . . . . 7  |-  (RngCatOLD `  U
)  =  C
3130fveq2i 5875 . . . . . 6  |-  ( Base `  (RngCatOLD `  U )
)  =  ( Base `  C )
3229, 31, 9rngcbasOLD 32935 . . . . 5  |-  ( ph  ->  ( Base `  (RngCatOLD `  U ) )  =  ( U  i^i Rng )
)
3332adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  R )  ->  ( Base `  (RngCatOLD `  U
) )  =  ( U  i^i Rng ) )
3428, 33eleqtrrd 2548 . . 3  |-  ( (
ph  /\  x  e.  R )  ->  x  e.  ( Base `  (RngCatOLD `  U ) ) )
35 fvex 5882 . . . . 5  |-  ( Base `  x )  e.  _V
3635a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  R )  ->  ( Base `  x )  e. 
_V )
3736resiexd 6139 . . 3  |-  ( (
ph  /\  x  e.  R )  ->  (  _I  |`  ( Base `  x
) )  e.  _V )
3815, 18, 34, 37fvmptd 5961 . 2  |-  ( (
ph  /\  x  e.  R )  ->  (
( Id `  (RngCatOLD `  U ) ) `  x )  =  (  _I  |`  ( Base `  x ) ) )
39 rngcrescrhmOLD.h . . . 4  |-  H  =  ( RingHom  |`  ( R  X.  R ) )
409, 29, 1, 39rhmsubcOLDlem2 33058 . . 3  |-  ( (
ph  /\  x  e.  R  /\  x  e.  R
)  ->  ( x H x )  =  ( x RingHom  x ) )
41403anidm23 1287 . 2  |-  ( (
ph  /\  x  e.  R )  ->  (
x H x )  =  ( x RingHom  x
) )
428, 38, 413eltr4d 2560 1  |-  ( (
ph  /\  x  e.  R )  ->  (
( Id `  (RngCatOLD `  U ) ) `  x )  e.  ( x H x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    |-> cmpt 4515    _I cid 4799    X. cxp 5006    |` cres 5010   ` cfv 5594  (class class class)co 6296   Basecbs 14644   Catccat 15081   Idccid 15082   Ringcrg 17325   RingHom crh 17488  Rngcrng 32824  RngCatOLDcrngcOLD 32910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-hom 14736  df-cco 14737  df-0g 14859  df-cat 15085  df-cid 15086  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-grp 16184  df-minusg 16185  df-ghm 16392  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-rnghom 17491  df-mgmhm 32729  df-rng0 32825  df-rnghomo 32837  df-rngcOLD 32912
This theorem is referenced by:  rhmsubcOLD  33061
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