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| Mirrors > Home > MPE Home > Th. List > 01eq0ringOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of 01eq0ring 20531 as of 23-Feb-2025. (Contributed by AV, 16-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| 0ring.b | ⊢ 𝐵 = (Base‘𝑅) | 
| 0ring.0 | ⊢ 0 = (0g‘𝑅) | 
| 0ring01eq.1 | ⊢ 1 = (1r‘𝑅) | 
| Ref | Expression | 
|---|---|
| 01eq0ringOLD | ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0ring.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | 1 | fvexi 6919 | . . . . . 6 ⊢ 𝐵 ∈ V | 
| 3 | hashv01gt1 14385 | . . . . . 6 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))) | |
| 4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) | 
| 5 | hasheq0 14403 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅)) | |
| 6 | 2, 5 | ax-mp 5 | . . . . . . . 8 ⊢ ((♯‘𝐵) = 0 ↔ 𝐵 = ∅) | 
| 7 | ne0i 4340 | . . . . . . . . 9 ⊢ ( 0 ∈ 𝐵 → 𝐵 ≠ ∅) | |
| 8 | eqneqall 2950 | . . . . . . . . 9 ⊢ (𝐵 = ∅ → (𝐵 ≠ ∅ → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | |
| 9 | 7, 8 | syl5com 31 | . . . . . . . 8 ⊢ ( 0 ∈ 𝐵 → (𝐵 = ∅ → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | 
| 10 | 6, 9 | biimtrid 242 | . . . . . . 7 ⊢ ( 0 ∈ 𝐵 → ((♯‘𝐵) = 0 → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | 
| 11 | 0ring.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
| 12 | 1, 11 | ring0cl 20265 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) | 
| 13 | 10, 12 | syl11 33 | . . . . . 6 ⊢ ((♯‘𝐵) = 0 → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | 
| 14 | eqneqall 2950 | . . . . . . 7 ⊢ ((♯‘𝐵) = 1 → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 )) | |
| 15 | 14 | a1d 25 | . . . . . 6 ⊢ ((♯‘𝐵) = 1 → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | 
| 16 | 0ring01eq.1 | . . . . . . . . . . 11 ⊢ 1 = (1r‘𝑅) | |
| 17 | 1, 16, 11 | ring1ne0 20297 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 ) | 
| 18 | 17 | necomd 2995 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 0 ≠ 1 ) | 
| 19 | 18 | ex 412 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1 < (♯‘𝐵) → 0 ≠ 1 )) | 
| 20 | 19 | a1i 11 | . . . . . . 7 ⊢ ((♯‘𝐵) ≠ 1 → (𝑅 ∈ Ring → (1 < (♯‘𝐵) → 0 ≠ 1 ))) | 
| 21 | 20 | com13 88 | . . . . . 6 ⊢ (1 < (♯‘𝐵) → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | 
| 22 | 13, 15, 21 | 3jaoi 1429 | . . . . 5 ⊢ (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) | 
| 23 | 4, 22 | ax-mp 5 | . . . 4 ⊢ (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 )) | 
| 24 | 23 | necon4d 2963 | . . 3 ⊢ (𝑅 ∈ Ring → ( 0 = 1 → (♯‘𝐵) = 1)) | 
| 25 | 24 | imp 406 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → (♯‘𝐵) = 1) | 
| 26 | 1, 11 | 0ring 20527 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) | 
| 27 | 25, 26 | syldan 591 | 1 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∅c0 4332 {csn 4625 class class class wbr 5142 ‘cfv 6560 0cc0 11156 1c1 11157 < clt 11296 ♯chash 14370 Basecbs 17248 0gc0g 17485 1rcur 20179 Ringcrg 20231 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-hash 14371 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 | 
| This theorem is referenced by: (None) | 
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