frgpcpbl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > frgpcpbl		Structured version Visualization version GIF version

Theorem frgpcpbl 19679

Description: Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)

**Hypotheses**
Ref	Expression
frgpval.m	⊢ 𝐺 = (freeGrp‘𝐼)
frgpval.b	⊢ 𝑀 = (freeMnd‘(𝐼 × 2_o))
frgpval.r	⊢ ∼ = ( ~_FG ‘𝐼)
frgpcpbl.p	⊢ + = (+_g‘𝑀)

**Assertion**
Ref	Expression
frgpcpbl	⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷))

Proof of Theorem frgpcpbl Dummy variables 𝑘 𝑚 𝑛 𝑡 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2726	. . 3 ⊢ ( I ‘Word (𝐼 × 2_o)) = ( I ‘Word (𝐼 × 2_o))
2		frgpval.r	. . 3 ⊢ ∼ = ( ~_FG ‘𝐼)
3		eqid 2726	. . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
4		eqid 2726	. . 3 ⊢ (𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩))) = (𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))
5		eqid 2726	. . 3 ⊢ (( I ‘Word (𝐼 × 2_o)) ∖ ∪ 𝑥 ∈ ( I ‘Word (𝐼 × 2_o))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥)) = (( I ‘Word (𝐼 × 2_o)) ∖ ∪ 𝑥 ∈ ( I ‘Word (𝐼 × 2_o))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥))
6		eqid 2726	. . 3 ⊢ (𝑚 ∈ {𝑡 ∈ (Word ( I ‘Word (𝐼 × 2_o)) ∖ {∅}) ∣ ((𝑡‘0) ∈ (( I ‘Word (𝐼 × 2_o)) ∖ ∪ 𝑥 ∈ ( I ‘Word (𝐼 × 2_o))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥)) ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) = (𝑚 ∈ {𝑡 ∈ (Word ( I ‘Word (𝐼 × 2_o)) ∖ {∅}) ∣ ((𝑡‘0) ∈ (( I ‘Word (𝐼 × 2_o)) ∖ ∪ 𝑥 ∈ ( I ‘Word (𝐼 × 2_o))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥)) ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2_o)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
7	1, 2, 3, 4, 5, 6	efgcpbl2 19677	. 2 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 ++ 𝐵) ∼ (𝐶 ++ 𝐷))
8	1, 2	efger 19638	. . . . . 6 ⊢ ∼ Er ( I ‘Word (𝐼 × 2_o))
9	8	a1i 11	. . . . 5 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → ∼ Er ( I ‘Word (𝐼 × 2_o)))
10		simpl 482	. . . . 5 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐴 ∼ 𝐶)
11	9, 10	ercl 8716	. . . 4 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐴 ∈ ( I ‘Word (𝐼 × 2_o)))
12	1	efgrcl 19635	. . . . . . 7 ⊢ (𝐴 ∈ ( I ‘Word (𝐼 × 2_o)) → (𝐼 ∈ V ∧ ( I ‘Word (𝐼 × 2_o)) = Word (𝐼 × 2_o)))
13	11, 12	syl 17	. . . . . 6 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐼 ∈ V ∧ ( I ‘Word (𝐼 × 2_o)) = Word (𝐼 × 2_o)))
14	13	simprd 495	. . . . 5 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → ( I ‘Word (𝐼 × 2_o)) = Word (𝐼 × 2_o))
15	13	simpld 494	. . . . . . 7 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐼 ∈ V)
16		2on 8481	. . . . . . 7 ⊢ 2_o ∈ On
17		xpexg 7734	. . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 2_o ∈ On) → (𝐼 × 2_o) ∈ V)
18	15, 16, 17	sylancl 585	. . . . . 6 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐼 × 2_o) ∈ V)
19		frgpval.b	. . . . . . 7 ⊢ 𝑀 = (freeMnd‘(𝐼 × 2_o))
20		eqid 2726	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
21	19, 20	frmdbas 18777	. . . . . 6 ⊢ ((𝐼 × 2_o) ∈ V → (Base‘𝑀) = Word (𝐼 × 2_o))
22	18, 21	syl 17	. . . . 5 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (Base‘𝑀) = Word (𝐼 × 2_o))
23	14, 22	eqtr4d 2769	. . . 4 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → ( I ‘Word (𝐼 × 2_o)) = (Base‘𝑀))
24	11, 23	eleqtrd 2829	. . 3 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐴 ∈ (Base‘𝑀))
25		simpr 484	. . . . 5 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐵 ∼ 𝐷)
26	9, 25	ercl 8716	. . . 4 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐵 ∈ ( I ‘Word (𝐼 × 2_o)))
27	26, 23	eleqtrd 2829	. . 3 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐵 ∈ (Base‘𝑀))
28		frgpcpbl.p	. . . 4 ⊢ + = (+_g‘𝑀)
29	19, 20, 28	frmdadd 18780	. . 3 ⊢ ((𝐴 ∈ (Base‘𝑀) ∧ 𝐵 ∈ (Base‘𝑀)) → (𝐴 + 𝐵) = (𝐴 ++ 𝐵))
30	24, 27, 29	syl2anc 583	. 2 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) = (𝐴 ++ 𝐵))
31	9, 10	ercl2 8718	. . . 4 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐶 ∈ ( I ‘Word (𝐼 × 2_o)))
32	31, 23	eleqtrd 2829	. . 3 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐶 ∈ (Base‘𝑀))
33	9, 25	ercl2 8718	. . . 4 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐷 ∈ ( I ‘Word (𝐼 × 2_o)))
34	33, 23	eleqtrd 2829	. . 3 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → 𝐷 ∈ (Base‘𝑀))
35	19, 20, 28	frmdadd 18780	. . 3 ⊢ ((𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 + 𝐷) = (𝐶 ++ 𝐷))
36	32, 34, 35	syl2anc 583	. 2 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐶 + 𝐷) = (𝐶 ++ 𝐷))
37	7, 30, 36	3brtr4d 5173	1 ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 {crab 3426 Vcvv 3468 ∖ cdif 3940 ∅c0 4317 {csn 4623 ⟨cop 4629 ⟨cotp 4631 ∪ ciun 4990 class class class wbr 5141 ↦ cmpt 5224 I cid 5566 × cxp 5667 ran crn 5670 Oncon0 6358 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 1_oc1o 8460 2_oc2o 8461 Er wer 8702 0cc0 11112 1c1 11113 − cmin 11448 ...cfz 13490 ..^cfzo 13633 ♯chash 14295 Word cword 14470 ++ cconcat 14526 splice csplice 14705 ⟨“cs2 14798 Basecbs 17153 +_gcplusg 17206 freeMndcfrmd 18772 ~_FG cefg 19626 freeGrpcfrgp 19627

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-ec 8707 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-concat 14527 df-s1 14552 df-substr 14597 df-pfx 14627 df-splice 14706 df-s2 14805 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-frmd 18774 df-efg 19629

This theorem is referenced by: frgp0 19680 frgpadd 19683

W3C validator