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Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetclem7ALTV | Structured version Visualization version GIF version |
Description: Lemma 7 for funcringcsetcALTV 44692. (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 44687 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 RingHom 𝑋))) |
9 | 8 | anabsan2 673 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 RingHom 𝑋))) |
10 | eqid 2798 | . . . 4 ⊢ (Id‘𝑅) = (Id‘𝑅) | |
11 | 5 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑈 ∈ WUni) |
12 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
13 | eqid 2798 | . . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
14 | 1, 3, 10, 11, 12, 13 | ringcidALTV 44678 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑅)‘𝑋) = ( I ↾ (Base‘𝑋))) |
15 | 9, 14 | fveq12d 6652 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋)))) |
16 | 1, 3, 5 | ringcbasALTV 44670 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
17 | 16 | eleq2d 2875 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Ring))) |
18 | elin 3897 | . . . . . 6 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring)) | |
19 | 18 | simprbi 500 | . . . . 5 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋 ∈ Ring) |
20 | 17, 19 | syl6bi 256 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → 𝑋 ∈ Ring)) |
21 | 20 | imp 410 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ Ring) |
22 | 13 | idrhm 19479 | . . 3 ⊢ (𝑋 ∈ Ring → ( I ↾ (Base‘𝑋)) ∈ (𝑋 RingHom 𝑋)) |
23 | fvresi 6912 | . . 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 44683 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) |
26 | 25 | fveq2d 6649 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(𝐹‘𝑋)) = ((Id‘𝑆)‘(Base‘𝑋))) |
27 | eqid 2798 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
28 | 1, 3, 5 | ringcbasbasALTV 44682 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ 𝑈) |
29 | 2, 27, 11, 28 | setcid 17338 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋))) |
30 | 26, 29 | eqtr2d 2834 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
31 | 15, 24, 30 | 3eqtrd 2837 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ↦ cmpt 5110 I cid 5424 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 WUnicwun 10111 Basecbs 16475 Idccid 16928 SetCatcsetc 17327 Ringcrg 19290 RingHom crh 19460 RingCatALTVcringcALTV 44628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-wun 10113 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 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 19233 df-ur 19245 df-ring 19292 df-rnghom 19463 df-ringcALTV 44630 |
This theorem is referenced by: funcringcsetcALTV 44692 |
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