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Mirrors > Home > MPE Home > Th. List > f1rhm0to0ALT | Structured version Visualization version GIF version |
Description: Alternate proof for f1rhm0to0 18740. Using ghmf1 17689 does not make the proof shorter and requires disjoint variable restrictions! (Contributed by AV, 24-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
f1rhm0to0.a | ⊢ 𝐴 = (Base‘𝑅) |
f1rhm0to0.b | ⊢ 𝐵 = (Base‘𝑆) |
f1rhm0to0.n | ⊢ 𝑁 = (0g‘𝑆) |
f1rhm0to0.0 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
f1rhm0to0ALT | ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmghm 18725 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
2 | 1 | adantr 481 | . . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑋 ∈ 𝐴) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) |
3 | f1rhm0to0.a | . . . . . . . 8 ⊢ 𝐴 = (Base‘𝑅) | |
4 | f1rhm0to0.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆) | |
5 | f1rhm0to0.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
6 | f1rhm0to0.n | . . . . . . . 8 ⊢ 𝑁 = (0g‘𝑆) | |
7 | 3, 4, 5, 6 | ghmf1 17689 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 𝑁 → 𝑥 = 0 ))) |
8 | 2, 7 | syl 17 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑋 ∈ 𝐴) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 𝑁 → 𝑥 = 0 ))) |
9 | fveq2 6191 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
10 | 9 | eqeq1d 2624 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑁 ↔ (𝐹‘𝑋) = 𝑁)) |
11 | eqeq1 2626 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0 )) | |
12 | 10, 11 | imbi12d 334 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑥) = 𝑁 → 𝑥 = 0 ) ↔ ((𝐹‘𝑋) = 𝑁 → 𝑋 = 0 ))) |
13 | 12 | rspcv 3305 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 𝑁 → 𝑥 = 0 ) → ((𝐹‘𝑋) = 𝑁 → 𝑋 = 0 ))) |
14 | 13 | adantl 482 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑋 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 𝑁 → 𝑥 = 0 ) → ((𝐹‘𝑋) = 𝑁 → 𝑋 = 0 ))) |
15 | 8, 14 | sylbid 230 | . . . . 5 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑋 ∈ 𝐴) → (𝐹:𝐴–1-1→𝐵 → ((𝐹‘𝑋) = 𝑁 → 𝑋 = 0 ))) |
16 | 15 | ex 450 | . . . 4 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑋 ∈ 𝐴 → (𝐹:𝐴–1-1→𝐵 → ((𝐹‘𝑋) = 𝑁 → 𝑋 = 0 )))) |
17 | 16 | com23 86 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐴–1-1→𝐵 → (𝑋 ∈ 𝐴 → ((𝐹‘𝑋) = 𝑁 → 𝑋 = 0 )))) |
18 | 17 | 3imp 1256 | . 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 → 𝑋 = 0 )) |
19 | fveq2 6191 | . . . 4 ⊢ (𝑋 = 0 → (𝐹‘𝑋) = (𝐹‘ 0 )) | |
20 | 5, 6 | ghmid 17666 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘ 0 ) = 𝑁) |
21 | 1, 20 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘ 0 ) = 𝑁) |
22 | 21 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘ 0 ) = 𝑁) |
23 | 19, 22 | sylan9eqr 2678 | . . 3 ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝐹‘𝑋) = 𝑁) |
24 | 23 | ex 450 | . 2 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → (𝑋 = 0 → (𝐹‘𝑋) = 𝑁)) |
25 | 18, 24 | impbid 202 | 1 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 0gc0g 16100 GrpHom cghm 17657 RingHom crh 18712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-ghm 17658 df-mgp 18490 df-ur 18502 df-ring 18549 df-rnghom 18715 |
This theorem is referenced by: (None) |
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