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Theorem funcringcsetclem7ALTV 44327
Description: Lemma 7 for funcringcsetcALTV 44330. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV.r 𝑅 = (RingCatALTV‘𝑈)
funcringcsetcALTV.s 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV.b 𝐵 = (Base‘𝑅)
funcringcsetcALTV.c 𝐶 = (Base‘𝑆)
funcringcsetcALTV.u (𝜑𝑈 ∈ WUni)
funcringcsetcALTV.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetclem7ALTV ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝜑,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcringcsetclem7ALTV
StepHypRef Expression
1 funcringcsetcALTV.r . . . . 5 𝑅 = (RingCatALTV‘𝑈)
2 funcringcsetcALTV.s . . . . 5 𝑆 = (SetCat‘𝑈)
3 funcringcsetcALTV.b . . . . 5 𝐵 = (Base‘𝑅)
4 funcringcsetcALTV.c . . . . 5 𝐶 = (Base‘𝑆)
5 funcringcsetcALTV.u . . . . 5 (𝜑𝑈 ∈ WUni)
6 funcringcsetcALTV.f . . . . 5 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
7 funcringcsetcALTV.g . . . . 5 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
81, 2, 3, 4, 5, 6, 7funcringcsetclem5ALTV 44325 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑋𝐵)) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 RingHom 𝑋)))
98anabsan2 672 . . 3 ((𝜑𝑋𝐵) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 RingHom 𝑋)))
10 eqid 2819 . . . 4 (Id‘𝑅) = (Id‘𝑅)
115adantr 483 . . . 4 ((𝜑𝑋𝐵) → 𝑈 ∈ WUni)
12 simpr 487 . . . 4 ((𝜑𝑋𝐵) → 𝑋𝐵)
13 eqid 2819 . . . 4 (Base‘𝑋) = (Base‘𝑋)
141, 3, 10, 11, 12, 13ringcidALTV 44316 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝑅)‘𝑋) = ( I ↾ (Base‘𝑋)))
159, 14fveq12d 6670 . 2 ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋))))
161, 3, 5ringcbasALTV 44308 . . . . . 6 (𝜑𝐵 = (𝑈 ∩ Ring))
1716eleq2d 2896 . . . . 5 (𝜑 → (𝑋𝐵𝑋 ∈ (𝑈 ∩ Ring)))
18 elin 4167 . . . . . 6 (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋𝑈𝑋 ∈ Ring))
1918simprbi 499 . . . . 5 (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋 ∈ Ring)
2017, 19syl6bi 255 . . . 4 (𝜑 → (𝑋𝐵𝑋 ∈ Ring))
2120imp 409 . . 3 ((𝜑𝑋𝐵) → 𝑋 ∈ Ring)
2213idrhm 19475 . . 3 (𝑋 ∈ Ring → ( I ↾ (Base‘𝑋)) ∈ (𝑋 RingHom 𝑋))
23 fvresi 6928 . . 3 (( I ↾ (Base‘𝑋)) ∈ (𝑋 RingHom 𝑋) → (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋))) = ( I ↾ (Base‘𝑋)))
2421, 22, 233syl 18 . 2 ((𝜑𝑋𝐵) → (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋))) = ( I ↾ (Base‘𝑋)))
251, 2, 3, 4, 5, 6funcringcsetclem1ALTV 44321 . . . 4 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
2625fveq2d 6667 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝑆)‘(𝐹𝑋)) = ((Id‘𝑆)‘(Base‘𝑋)))
27 eqid 2819 . . . 4 (Id‘𝑆) = (Id‘𝑆)
281, 3, 5ringcbasbasALTV 44320 . . . 4 ((𝜑𝑋𝐵) → (Base‘𝑋) ∈ 𝑈)
292, 27, 11, 28setcid 17338 . . 3 ((𝜑𝑋𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋)))
3026, 29eqtr2d 2855 . 2 ((𝜑𝑋𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
3115, 24, 303eqtrd 2858 1 ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  cin 3933  cmpt 5137   I cid 5452  cres 5550  cfv 6348  (class class class)co 7148  cmpo 7150  WUnicwun 10114  Basecbs 16475  Idccid 16928  SetCatcsetc 17327  Ringcrg 19289   RingHom crh 19456  RingCatALTVcringcALTV 44266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-wun 10116  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-hom 16581  df-cco 16582  df-0g 16707  df-cat 16931  df-cid 16932  df-setc 17328  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-grp 18098  df-ghm 18348  df-mgp 19232  df-ur 19244  df-ring 19291  df-rnghom 19459  df-ringcALTV 44268
This theorem is referenced by:  funcringcsetcALTV  44330
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