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Mirrors > Home > MPE Home > Th. List > symgfvne | Structured version Visualization version GIF version |
Description: The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019.) |
Ref | Expression |
---|---|
symgbas.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symgbas.2 | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
symgfvne | ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgbas.1 | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | symgbas.2 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | symgbasf1o 18439 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴) |
4 | f1of1 6607 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴–1-1→𝐴) | |
5 | eqeq2 2830 | . . . . . . . 8 ⊢ (𝑍 = (𝐹‘𝑋) → ((𝐹‘𝑌) = 𝑍 ↔ (𝐹‘𝑌) = (𝐹‘𝑋))) | |
6 | 5 | eqcoms 2826 | . . . . . . 7 ⊢ ((𝐹‘𝑋) = 𝑍 → ((𝐹‘𝑌) = 𝑍 ↔ (𝐹‘𝑌) = (𝐹‘𝑋))) |
7 | 6 | adantl 482 | . . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → ((𝐹‘𝑌) = 𝑍 ↔ (𝐹‘𝑌) = (𝐹‘𝑋))) |
8 | simp1 1128 | . . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐴) | |
9 | simp3 1130 | . . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐴) | |
10 | simp2 1129 | . . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
11 | f1veqaeq 7006 | . . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ (𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋)) | |
12 | 8, 9, 10, 11 | syl12anc 832 | . . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋)) |
13 | 12 | adantr 481 | . . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋)) |
14 | 7, 13 | sylbid 241 | . . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → ((𝐹‘𝑌) = 𝑍 → 𝑌 = 𝑋)) |
15 | 14 | necon3d 3034 | . . . 4 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍)) |
16 | 15 | 3exp1 1344 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐴 → (𝑋 ∈ 𝐴 → (𝑌 ∈ 𝐴 → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍))))) |
17 | 3, 4, 16 | 3syl 18 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝑋 ∈ 𝐴 → (𝑌 ∈ 𝐴 → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍))))) |
18 | 17 | 3imp 1103 | 1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ‘cfv 6348 Basecbs 16471 SymGrpcsymg 18433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-tset 16572 df-symg 18434 |
This theorem is referenced by: gsummatr01lem4 21195 |
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