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Mirrors > Home > MPE Home > Th. List > symgextf1lem | Structured version Visualization version GIF version |
Description: Lemma for symgextf1 19414. (Contributed by AV, 6-Jan-2019.) |
Ref | Expression |
---|---|
symgext.s | ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) |
symgext.e | ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥))) |
Ref | Expression |
---|---|
symgextf1lem | ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾}) → (𝐸‘𝑋) ≠ (𝐸‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . . . . 7 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾})) | |
2 | symgext.s | . . . . . . 7 ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) | |
3 | 1, 2 | symgfv 19372 | . . . . . 6 ⊢ ((𝑍 ∈ 𝑆 ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝑍‘𝑋) ∈ (𝑁 ∖ {𝐾})) |
4 | 3 | adantll 712 | . . . . 5 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝑍‘𝑋) ∈ (𝑁 ∖ {𝐾})) |
5 | eldifsni 4798 | . . . . . 6 ⊢ ((𝑍‘𝑋) ∈ (𝑁 ∖ {𝐾}) → (𝑍‘𝑋) ≠ 𝐾) | |
6 | symgext.e | . . . . . . . . 9 ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥))) | |
7 | 2, 6 | symgextfv 19411 | . . . . . . . 8 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸‘𝑋) = (𝑍‘𝑋))) |
8 | 7 | imp 405 | . . . . . . 7 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸‘𝑋) = (𝑍‘𝑋)) |
9 | 8 | neeq1d 2989 | . . . . . 6 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → ((𝐸‘𝑋) ≠ 𝐾 ↔ (𝑍‘𝑋) ≠ 𝐾)) |
10 | 5, 9 | imbitrrid 245 | . . . . 5 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → ((𝑍‘𝑋) ∈ (𝑁 ∖ {𝐾}) → (𝐸‘𝑋) ≠ 𝐾)) |
11 | 4, 10 | mpd 15 | . . . 4 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸‘𝑋) ≠ 𝐾) |
12 | 11 | adantrr 715 | . . 3 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ (𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾})) → (𝐸‘𝑋) ≠ 𝐾) |
13 | elsni 4649 | . . . . . 6 ⊢ (𝑌 ∈ {𝐾} → 𝑌 = 𝐾) | |
14 | 2, 6 | symgextfve 19412 | . . . . . . 7 ⊢ (𝐾 ∈ 𝑁 → (𝑌 = 𝐾 → (𝐸‘𝑌) = 𝐾)) |
15 | 14 | adantr 479 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝑌 = 𝐾 → (𝐸‘𝑌) = 𝐾)) |
16 | 13, 15 | syl5com 31 | . . . . 5 ⊢ (𝑌 ∈ {𝐾} → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝐸‘𝑌) = 𝐾)) |
17 | 16 | adantl 480 | . . . 4 ⊢ ((𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾}) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝐸‘𝑌) = 𝐾)) |
18 | 17 | impcom 406 | . . 3 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ (𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾})) → (𝐸‘𝑌) = 𝐾) |
19 | 12, 18 | neeqtrrd 3004 | . 2 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ (𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾})) → (𝐸‘𝑋) ≠ (𝐸‘𝑌)) |
20 | 19 | ex 411 | 1 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾}) → (𝐸‘𝑋) ≠ (𝐸‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3943 ifcif 4532 {csn 4632 ↦ cmpt 5235 ‘cfv 6553 Basecbs 17208 SymGrpcsymg 19359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-uz 12870 df-fz 13534 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-tset 17280 df-efmnd 18854 df-symg 19360 |
This theorem is referenced by: symgextf1 19414 |
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