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Mirrors > Home > MPE Home > Th. List > efgi0 | Structured version Visualization version GIF version |
Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
efgval.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) |
efgval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
Ref | Expression |
---|---|
efgi0 | ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5308 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | 1 | prid1 4768 | . . . . 5 ⊢ ∅ ∈ {∅, 1o} |
3 | df2o3 8495 | . . . . 5 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2824 | . . . 4 ⊢ ∅ ∈ 2o |
5 | efgval.w | . . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
6 | efgval.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
7 | 5, 6 | efgi 19686 | . . . 4 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ ∅ ∈ 2o)) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉)) |
8 | 4, 7 | mpanr2 702 | . . 3 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉)) |
9 | 8 | 3impa 1107 | . 2 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉)) |
10 | tru 1537 | . . . 4 ⊢ ⊤ | |
11 | eqidd 2726 | . . . . 5 ⊢ (⊤ → 〈𝐽, ∅〉 = 〈𝐽, ∅〉) | |
12 | dif0 4374 | . . . . . . 7 ⊢ (1o ∖ ∅) = 1o | |
13 | 12 | opeq2i 4879 | . . . . . 6 ⊢ 〈𝐽, (1o ∖ ∅)〉 = 〈𝐽, 1o〉 |
14 | 13 | a1i 11 | . . . . 5 ⊢ (⊤ → 〈𝐽, (1o ∖ ∅)〉 = 〈𝐽, 1o〉) |
15 | 11, 14 | s2eqd 14850 | . . . 4 ⊢ (⊤ → 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉 = 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉) |
16 | oteq3 4886 | . . . 4 ⊢ (〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉 = 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉 → 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉 = 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉) | |
17 | 10, 15, 16 | mp2b 10 | . . 3 ⊢ 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉 = 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉 |
18 | 17 | oveq2i 7430 | . 2 ⊢ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉) = (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉) |
19 | 9, 18 | breqtrdi 5190 | 1 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∖ cdif 3941 ∅c0 4322 {cpr 4632 〈cop 4636 〈cotp 4638 class class class wbr 5149 I cid 5575 × cxp 5676 ‘cfv 6549 (class class class)co 7419 1oc1o 8480 2oc2o 8481 0cc0 11140 ...cfz 13519 ♯chash 14325 Word cword 14500 splice csplice 14735 〈“cs2 14828 ~FG cefg 19673 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
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 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-concat 14557 df-s1 14582 df-substr 14627 df-pfx 14657 df-splice 14736 df-s2 14835 df-efg 19676 |
This theorem is referenced by: (None) |
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