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Mirrors > Home > MPE Home > Th. List > 01eq0ringOLD | Structured version Visualization version GIF version |
Description: Obsolete version of 01eq0ring 20556 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 6934 | . . . . . 6 ⊢ 𝐵 ∈ V |
3 | hashv01gt1 14394 | . . . . . 6 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) |
5 | hasheq0 14412 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅)) | |
6 | 2, 5 | ax-mp 5 | . . . . . . . 8 ⊢ ((♯‘𝐵) = 0 ↔ 𝐵 = ∅) |
7 | ne0i 4364 | . . . . . . . . 9 ⊢ ( 0 ∈ 𝐵 → 𝐵 ≠ ∅) | |
8 | eqneqall 2957 | . . . . . . . . 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 20290 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
13 | 10, 12 | syl11 33 | . . . . . 6 ⊢ ((♯‘𝐵) = 0 → (𝑅 ∈ Ring → ((♯‘𝐵) ≠ 1 → 0 ≠ 1 ))) |
14 | eqneqall 2957 | . . . . . . 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 20322 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 ) |
18 | 17 | necomd 3002 | . . . . . . . . 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 1428 | . . . . 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 2970 | . . 3 ⊢ (𝑅 ∈ Ring → ( 0 = 1 → (♯‘𝐵) = 1)) |
25 | 24 | imp 406 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → (♯‘𝐵) = 1) |
26 | 1, 11 | 0ring 20552 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 }) |
27 | 25, 26 | syldan 590 | 1 ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1086 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ∅c0 4352 {csn 4648 class class class wbr 5166 ‘cfv 6573 0cc0 11184 1c1 11185 < clt 11324 ♯chash 14379 Basecbs 17258 0gc0g 17499 1rcur 20208 Ringcrg 20260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-fz 13568 df-hash 14380 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-0g 17501 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 |
This theorem is referenced by: (None) |
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