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Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetclem7ALTV | Structured version Visualization version GIF version |
Description: Lemma 7 for funcringcsetcALTV 44333. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
funcringcsetcALTV.r | ⊢ 𝑅 = (RingCatALTV‘𝑈) |
funcringcsetcALTV.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcringcsetcALTV.b | ⊢ 𝐵 = (Base‘𝑅) |
funcringcsetcALTV.c | ⊢ 𝐶 = (Base‘𝑆) |
funcringcsetcALTV.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcringcsetcALTV.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) |
funcringcsetcALTV.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) |
Ref | Expression |
---|---|
funcringcsetclem7ALTV | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
Step | Hyp | Ref | 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 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | funcringcsetcALTV.f | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) | |
7 | funcringcsetcALTV.g | . . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | funcringcsetclem5ALTV 44328 | . . . 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 | ringcidALTV 44319 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑅)‘𝑋) = ( I ↾ (Base‘𝑋))) |
15 | 9, 14 | fveq12d 6671 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋)))) |
16 | 1, 3, 5 | ringcbasALTV 44311 | . . . . . 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 19477 | . . 3 ⊢ (𝑋 ∈ Ring → ( I ↾ (Base‘𝑋)) ∈ (𝑋 RingHom 𝑋)) |
23 | fvresi 6929 | . . 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 | funcringcsetclem1ALTV 44324 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) |
26 | 25 | fveq2d 6668 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(𝐹‘𝑋)) = ((Id‘𝑆)‘(Base‘𝑋))) |
27 | eqid 2821 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
28 | 1, 3, 5 | ringcbasbasALTV 44323 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ 𝑈) |
29 | 2, 27, 11, 28 | setcid 17340 | . . 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 5138 I cid 5453 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 WUnicwun 10116 Basecbs 16477 Idccid 16930 SetCatcsetc 17329 Ringcrg 19291 RingHom crh 19458 RingCatALTVcringcALTV 44269 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-wun 10118 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-hom 16583 df-cco 16584 df-0g 16709 df-cat 16933 df-cid 16934 df-setc 17330 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-grp 18100 df-ghm 18350 df-mgp 19234 df-ur 19246 df-ring 19293 df-rnghom 19461 df-ringcALTV 44271 |
This theorem is referenced by: funcringcsetcALTV 44333 |
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