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Mirrors > Home > MPE Home > Th. List > symgextf1lem | Structured version Visualization version GIF version |
Description: Lemma for symgextf1 18552. (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 2824 | . . . . . . 7 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾})) | |
2 | symgext.s | . . . . . . 7 ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) | |
3 | 1, 2 | symgfv 18511 | . . . . . 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 4725 | . . . . . 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 18549 | . . . . . . . 8 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸‘𝑋) = (𝑍‘𝑋))) |
8 | 7 | imp 409 | . . . . . . 7 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸‘𝑋) = (𝑍‘𝑋)) |
9 | 8 | neeq1d 3078 | . . . . . 6 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → ((𝐸‘𝑋) ≠ 𝐾 ↔ (𝑍‘𝑋) ≠ 𝐾)) |
10 | 5, 9 | syl5ibr 248 | . . . . 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 4587 | . . . . . 6 ⊢ (𝑌 ∈ {𝐾} → 𝑌 = 𝐾) | |
14 | 2, 6 | symgextfve 18550 | . . . . . . 7 ⊢ (𝐾 ∈ 𝑁 → (𝑌 = 𝐾 → (𝐸‘𝑌) = 𝐾)) |
15 | 14 | adantr 483 | . . . . . 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 484 | . . . 4 ⊢ ((𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾}) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝐸‘𝑌) = 𝐾)) |
18 | 17 | impcom 410 | . . 3 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ (𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾})) → (𝐸‘𝑌) = 𝐾) |
19 | 12, 18 | neeqtrrd 3093 | . 2 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ (𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾})) → (𝐸‘𝑋) ≠ (𝐸‘𝑌)) |
20 | 19 | ex 415 | 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 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∖ cdif 3936 ifcif 4470 {csn 4570 ↦ cmpt 5149 ‘cfv 6358 Basecbs 16486 SymGrpcsymg 18498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-tset 16587 df-efmnd 18037 df-symg 18499 |
This theorem is referenced by: symgextf1 18552 |
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