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Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetcALTV2lem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for funcringcsetcALTV2 41837. (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 41832 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 RingHom 𝑋))) |
9 | 8 | anabsan2 859 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝑋𝐺𝑋) = ( I ↾ (𝑋 RingHom 𝑋))) |
10 | eqid 2610 | . . . 4 ⊢ (Id‘𝑅) = (Id‘𝑅) | |
11 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑈 ∈ WUni) |
12 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
13 | eqid 2610 | . . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
14 | 1, 3, 10, 11, 12, 13 | ringcid 41817 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑅)‘𝑋) = ( I ↾ (Base‘𝑋))) |
15 | 9, 14 | fveq12d 6109 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = (( I ↾ (𝑋 RingHom 𝑋))‘( I ↾ (Base‘𝑋)))) |
16 | 1, 3, 5 | ringcbas 41803 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
17 | 16 | eleq2d 2673 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Ring))) |
18 | elin 3758 | . . . . . 6 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring)) | |
19 | 18 | simprbi 479 | . . . . 5 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋 ∈ Ring) |
20 | 17, 19 | syl6bi 242 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → 𝑋 ∈ Ring)) |
21 | 20 | imp 444 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ Ring) |
22 | 13 | idrhm 18554 | . . 3 ⊢ (𝑋 ∈ Ring → ( I ↾ (Base‘𝑋)) ∈ (𝑋 RingHom 𝑋)) |
23 | fvresi 6344 | . . 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 41828 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) |
26 | 25 | fveq2d 6107 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(𝐹‘𝑋)) = ((Id‘𝑆)‘(Base‘𝑋))) |
27 | eqid 2610 | . . . 4 ⊢ (Id‘𝑆) = (Id‘𝑆) | |
28 | 1, 3, 5 | ringcbasbas 41826 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ 𝑈) |
29 | 2, 27, 11, 28 | setcid 16559 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((Id‘𝑆)‘(Base‘𝑋)) = ( I ↾ (Base‘𝑋))) |
30 | 26, 29 | eqtr2d 2645 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ( I ↾ (Base‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
31 | 15, 24, 30 | 3eqtrd 2648 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ↦ cmpt 4643 I cid 4948 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 WUnicwun 9401 Basecbs 15695 Idccid 16149 SetCatcsetc 16548 Ringcrg 18370 RingHom crh 18535 RingCatcringc 41795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-wun 9403 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-hom 15793 df-cco 15794 df-0g 15925 df-cat 16152 df-cid 16153 df-homf 16154 df-ssc 16293 df-resc 16294 df-subc 16295 df-setc 16549 df-estrc 16586 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-ghm 17481 df-mgp 18313 df-ur 18325 df-ring 18372 df-rnghom 18538 df-ringc 41797 |
This theorem is referenced by: funcringcsetcALTV2 41837 |
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