Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetclem7ALTV | Structured version Visualization version GIF version |
Description: Lemma 7 for funcringcsetcALTV 44346. (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 44341 | . . . 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 2823 | . . . 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 2823 | . . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
14 | 1, 3, 10, 11, 12, 13 | ringcidALTV 44332 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑅)‘𝑋) = ( I ↾ (Base‘𝑋))) |
15 | 9, 14 | fveq12d 6679 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋)))) |
16 | 1, 3, 5 | ringcbasALTV 44324 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
17 | 16 | eleq2d 2900 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Ring))) |
18 | elin 4171 | . . . . . 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 19485 | . . 3 ⊢ (𝑋 ∈ Ring → ( I ↾ (Base‘𝑋)) ∈ (𝑋 RingHom 𝑋)) |
23 | fvresi 6937 | . . 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 44337 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) |
26 | 25 | fveq2d 6676 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(𝐹‘𝑋)) = ((Id‘𝑆)‘(Base‘𝑋))) |
27 | eqid 2823 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
28 | 1, 3, 5 | ringcbasbasALTV 44336 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ 𝑈) |
29 | 2, 27, 11, 28 | setcid 17348 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋))) |
30 | 26, 29 | eqtr2d 2859 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
31 | 15, 24, 30 | 3eqtrd 2862 | 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 1537 ∈ wcel 2114 ∩ cin 3937 ↦ cmpt 5148 I cid 5461 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 WUnicwun 10124 Basecbs 16485 Idccid 16938 SetCatcsetc 17337 Ringcrg 19299 RingHom crh 19466 RingCatALTVcringcALTV 44282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-wun 10126 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-hom 16591 df-cco 16592 df-0g 16717 df-cat 16941 df-cid 16942 df-setc 17338 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-ghm 18358 df-mgp 19242 df-ur 19254 df-ring 19301 df-rnghom 19469 df-ringcALTV 44284 |
This theorem is referenced by: funcringcsetcALTV 44346 |
Copyright terms: Public domain | W3C validator |