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Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetcALTV2lem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for funcringcsetcALTV2 44315. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
funcringcsetcALTV2.r | ⊢ 𝑅 = (RingCat‘𝑈) |
funcringcsetcALTV2.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcringcsetcALTV2.b | ⊢ 𝐵 = (Base‘𝑅) |
funcringcsetcALTV2.c | ⊢ 𝐶 = (Base‘𝑆) |
funcringcsetcALTV2.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcringcsetcALTV2.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) |
funcringcsetcALTV2.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) |
Ref | Expression |
---|---|
funcringcsetcALTV2lem7 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcringcsetcALTV2.r | . . . . 5 ⊢ 𝑅 = (RingCat‘𝑈) | |
2 | funcringcsetcALTV2.s | . . . . 5 ⊢ 𝑆 = (SetCat‘𝑈) | |
3 | funcringcsetcALTV2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
4 | funcringcsetcALTV2.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑆) | |
5 | funcringcsetcALTV2.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | funcringcsetcALTV2.f | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) | |
7 | funcringcsetcALTV2.g | . . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | funcringcsetcALTV2lem5 44310 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 RingHom 𝑋))) |
9 | 8 | anabsan2 672 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 RingHom 𝑋))) |
10 | eqid 2821 | . . . 4 ⊢ (Id‘𝑅) = (Id‘𝑅) | |
11 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑈 ∈ WUni) |
12 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
13 | eqid 2821 | . . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
14 | 1, 3, 10, 11, 12, 13 | ringcid 44295 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑅)‘𝑋) = ( I ↾ (Base‘𝑋))) |
15 | 9, 14 | fveq12d 6676 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋)))) |
16 | 1, 3, 5 | ringcbas 44281 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
17 | 16 | eleq2d 2898 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Ring))) |
18 | elin 4168 | . . . . . 6 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring)) | |
19 | 18 | simprbi 499 | . . . . 5 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋 ∈ Ring) |
20 | 17, 19 | syl6bi 255 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → 𝑋 ∈ Ring)) |
21 | 20 | imp 409 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ Ring) |
22 | 13 | idrhm 19482 | . . 3 ⊢ (𝑋 ∈ Ring → ( I ↾ (Base‘𝑋)) ∈ (𝑋 RingHom 𝑋)) |
23 | fvresi 6934 | . . 3 ⊢ (( I ↾ (Base‘𝑋)) ∈ (𝑋 RingHom 𝑋) → (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋))) = ( I ↾ (Base‘𝑋))) | |
24 | 21, 22, 23 | 3syl 18 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋))) = ( I ↾ (Base‘𝑋))) |
25 | 1, 2, 3, 4, 5, 6 | funcringcsetcALTV2lem1 44306 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) |
26 | 25 | fveq2d 6673 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(𝐹‘𝑋)) = ((Id‘𝑆)‘(Base‘𝑋))) |
27 | eqid 2821 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
28 | 1, 3, 5 | ringcbasbas 44304 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ 𝑈) |
29 | 2, 27, 11, 28 | setcid 17345 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋))) |
30 | 26, 29 | eqtr2d 2857 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
31 | 15, 24, 30 | 3eqtrd 2860 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ↦ cmpt 5145 I cid 5458 ↾ cres 5556 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 WUnicwun 10121 Basecbs 16482 Idccid 16935 SetCatcsetc 17334 Ringcrg 19296 RingHom crh 19463 RingCatcringc 44273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-wun 10123 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-hom 16588 df-cco 16589 df-0g 16714 df-cat 16938 df-cid 16939 df-homf 16940 df-ssc 17079 df-resc 17080 df-subc 17081 df-setc 17335 df-estrc 17372 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-grp 18105 df-ghm 18355 df-mgp 19239 df-ur 19251 df-ring 19298 df-rnghom 19466 df-ringc 44275 |
This theorem is referenced by: funcringcsetcALTV2 44315 |
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