![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > efgi1 | 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 |
---|---|
efgi1 | ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, ∅〉”〉〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8515 | . . . . . 6 ⊢ 1o ∈ V | |
2 | 1 | prid2 4768 | . . . . 5 ⊢ 1o ∈ {∅, 1o} |
3 | df2o3 8513 | . . . . 5 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2838 | . . . 4 ⊢ 1o ∈ 2o |
5 | efgval.w | . . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
6 | efgval.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
7 | 5, 6 | efgi 19752 | . . . 4 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 1o ∈ 2o)) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, (1o ∖ 1o)〉”〉〉)) |
8 | 4, 7 | mpanr2 704 | . . 3 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, (1o ∖ 1o)〉”〉〉)) |
9 | 8 | 3impa 1109 | . 2 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, (1o ∖ 1o)〉”〉〉)) |
10 | tru 1541 | . . . 4 ⊢ ⊤ | |
11 | eqidd 2736 | . . . . 5 ⊢ (⊤ → 〈𝐽, 1o〉 = 〈𝐽, 1o〉) | |
12 | difid 4382 | . . . . . . 7 ⊢ (1o ∖ 1o) = ∅ | |
13 | 12 | opeq2i 4882 | . . . . . 6 ⊢ 〈𝐽, (1o ∖ 1o)〉 = 〈𝐽, ∅〉 |
14 | 13 | a1i 11 | . . . . 5 ⊢ (⊤ → 〈𝐽, (1o ∖ 1o)〉 = 〈𝐽, ∅〉) |
15 | 11, 14 | s2eqd 14899 | . . . 4 ⊢ (⊤ → 〈“〈𝐽, 1o〉〈𝐽, (1o ∖ 1o)〉”〉 = 〈“〈𝐽, 1o〉〈𝐽, ∅〉”〉) |
16 | oteq3 4889 | . . . 4 ⊢ (〈“〈𝐽, 1o〉〈𝐽, (1o ∖ 1o)〉”〉 = 〈“〈𝐽, 1o〉〈𝐽, ∅〉”〉 → 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, (1o ∖ 1o)〉”〉〉 = 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, ∅〉”〉〉) | |
17 | 10, 15, 16 | mp2b 10 | . . 3 ⊢ 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, (1o ∖ 1o)〉”〉〉 = 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, ∅〉”〉〉 |
18 | 17 | oveq2i 7442 | . 2 ⊢ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, (1o ∖ 1o)〉”〉〉) = (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, ∅〉”〉〉) |
19 | 9, 18 | breqtrdi 5189 | 1 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, ∅〉”〉〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ∖ cdif 3960 ∅c0 4339 {cpr 4633 〈cop 4637 〈cotp 4639 class class class wbr 5148 I cid 5582 × cxp 5687 ‘cfv 6563 (class class class)co 7431 1oc1o 8498 2oc2o 8499 0cc0 11153 ...cfz 13544 ♯chash 14366 Word cword 14549 splice csplice 14784 〈“cs2 14877 ~FG cefg 19739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-s2 14884 df-efg 19742 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |