Theorem lmsslem 7949

Description: Lemma for lmss 7950 and causs 7952.

Hypotheses

Ref	Expression
lmsslem.1	|- (((D e. Met /\ ch) /\ ph) -> F (_ (CC X. dom dom D))
lmsslem.2	|- ((((D |` (Y X. Y)) e. Met /\ ch) /\ ps) -> F (_ (CC X. dom dom ( D |` (Y X. Y))))
lmsslem.3	|- ((D e. Met /\ (ch /\ F:NN-->dom dom ( D |` (Y X. Y)))) -> (ph <-> ps))
lmsslem.4	|- (ch -> F:NN-->Y)

Assertion

Ref	Expression
lmsslem	|- ((D e. Met /\ ch) -> (ph <-> ps))

Proof of Theorem lmsslem

Step	Hyp	Ref	Expression
1		lmsslem.1	. . . . 5 |- (((D e. Met /\ ch) /\ ph) -> F (_ (CC X. dom dom D))
2		lmsslem.2	. . . . . . 7 |- ((((D |` (Y X. Y)) e. Met /\ ch) /\ ps) -> F (_ (CC X. dom dom ( D |` (Y X. Y))))
3		metres 7820	. . . . . . 7 |- (D e. Met -> (D |` (Y X. Y)) e. Met)
4	2, 3	sylanl1 462	. . . . . 6 |- (((D e. Met /\ ch) /\ ps) -> F (_ (CC X. dom dom ( D |` (Y X. Y))))
5		ssid 2083	. . . . . . . 8 |- CC (_ CC
6		resss 3389	. . . . . . . . . 10 |- (D |` (Y X. Y)) (_ D
7		dmss 3316	. . . . . . . . . 10 |- ((D |` (Y X. Y)) (_ D -> dom ( D |` (Y X. Y)) (_ dom D)
8	6, 7	ax-mp 7	. . . . . . . . 9 |- dom ( D |` (Y X. Y)) (_ dom D
9		dmss 3316	. . . . . . . . 9 |- (dom ( D |` (Y X. Y)) (_ dom D -> dom dom ( D |` (Y X. Y)) (_ dom dom D)
10	8, 9	ax-mp 7	. . . . . . . 8 |- dom dom ( D |` (Y X. Y)) (_ dom dom D
11		ssxp 3262	. . . . . . . 8 |- ((CC (_ CC /\ dom dom ( D |` (Y X. Y)) (_ dom dom D) -> (CC X. dom dom ( D |` (Y X. Y))) (_ (CC X. dom dom D))
12	5, 10, 11	mp2an 699	. . . . . . 7 |- (CC X. dom dom ( D |` (Y X. Y))) (_ (CC X. dom dom D)
13		sstr 2075	. . . . . . 7 |- ((F (_ (CC X. dom dom ( D |` (Y X. Y))) /\ (CC X. dom dom ( D |` (Y X. Y))) (_ (CC X. dom dom D)) -> F (_ (CC X. dom dom D))
14	12, 13	mpan2 698	. . . . . 6 |- (F (_ (CC X. dom dom ( D |` (Y X. Y))) -> F (_ (CC X. dom dom D))
15	4, 14	syl 10	. . . . 5 |- (((D e. Met /\ ch) /\ ps) -> F (_ (CC X. dom dom D))
16	1, 15	jaodan 428	. . . 4 |- (((D e. Met /\ ch) /\ (ph \/ ps)) -> F (_ (CC X. dom dom D))
17	16	ex 373	. . 3 |- ((D e. Met /\ ch) -> ((ph \/ ps) -> F (_ (CC X. dom dom D)))
18		simprl 416	. . . . . . 7 |- ((D e. Met /\ (ch /\ F (_ (CC X. dom dom D))) -> ch)
19		eqid 1478	. . . . . . . . . . . . 13 |- dom dom D = dom dom D
20	19	metssba 7806	. . . . . . . . . . . 12 |- (D e. Met -> (dom dom D i^i Y) = dom dom ( D |` (Y X. Y)))
21		incom 2211	. . . . . . . . . . . 12 |- (Y i^i dom dom D) = (dom dom D i^i Y)
22	20, 21	syl5req 1523	. . . . . . . . . . 11 |- (D e. Met -> dom dom ( D |` (Y X. Y)) = (Y i^i dom dom D))
23		feq3 3628	. . . . . . . . . . 11 |- (dom dom ( D |` (Y X. Y)) = (Y i^i dom dom D) -> (F:NN-->dom dom ( D |` (Y X. Y)) <-> F:NN-->(Y i^i dom dom D)))
24	22, 23	syl 10	. . . . . . . . . 10 |- (D e. Met -> (F:NN-->dom dom ( D |` (Y X. Y)) <-> F:NN-->(Y i^i dom dom D)))
25		pm3.26 319	. . . . . . . . . . . 12 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> F:NN-->Y)
26		ffn 3633	. . . . . . . . . . . . . 14 |- (F:NN-->Y -> F Fn NN)
27		rnss 3348	. . . . . . . . . . . . . . 15 |- (F (_ (CC X. dom dom D) -> ran F (_ ran (CC X. dom dom D))
28		ax1cn 5281	. . . . . . . . . . . . . . . . 17 |- 1 e. CC
29		ne0i 2289	. . . . . . . . . . . . . . . . 17 |- (1 e. CC -> CC =/= (/))
30	28, 29	ax-mp 7	. . . . . . . . . . . . . . . 16 |- CC =/= (/)
31		rnxp 3478	. . . . . . . . . . . . . . . 16 |- (CC =/= (/) -> ran (CC X. dom dom D) = dom dom D)
32	30, 31	ax-mp 7	. . . . . . . . . . . . . . 15 |- ran (CC X. dom dom D) = dom dom D
33	27, 32	syl6ss 2110	. . . . . . . . . . . . . 14 |- (F (_ (CC X. dom dom D) -> ran F (_ dom dom D)
34	26, 33	anim12i 333	. . . . . . . . . . . . 13 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> (F Fn NN /\ ran F (_ dom dom D))
35		df-f 3200	. . . . . . . . . . . . 13 |- (F:NN-->dom dom D <-> (F Fn NN /\ ran F (_ dom dom D))
36	34, 35	sylibr 200	. . . . . . . . . . . 12 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> F:NN-->dom dom D)
37	25, 36	jca 288	. . . . . . . . . . 11 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> (F:NN-->Y /\ F:NN-->dom dom D))
38		fin 3657	. . . . . . . . . . 11 |- (F:NN-->(Y i^i dom dom D) <-> (F:NN-->Y /\ F:NN-->dom dom D))
39	37, 38	sylibr 200	. . . . . . . . . 10 |- ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> F:NN-->(Y i^i dom dom D))
40	24, 39	syl5bir 210	. . . . . . . . 9 |- (D e. Met -> ((F:NN-->Y /\ F (_ (CC X. dom dom D)) -> F:NN-->dom dom ( D |` (Y X. Y))))
41	40	imp 350	. . . . . . . 8 |- ((D e. Met /\ (F:NN-->Y /\ F (_ (CC X. dom dom D))) -> F:NN-->dom dom ( D |` (Y X. Y)))
42		lmsslem.4	. . . . . . . 8 |- (ch -> F:NN-->Y)
43	41, 42	sylanr1 464	. . . . . . 7 |- ((D e. Met /\ (ch /\ F (_ (CC X. dom dom D))) -> F:NN-->dom dom ( D |` (Y X. Y)))
44	18, 43	jca 288	. . . . . 6 |- ((D e. Met /\ (ch /\ F (_ (CC X. dom dom D))) -> (ch /\ F:NN-->dom dom ( D |` (Y X. Y))))
45		lmsslem.3	. . . . . 6 |- ((D e. Met /\ (ch /\ F:NN-->dom dom ( D |` (Y X. Y)))) -> (ph <-> ps))
46	44, 45	syldan 469	. . . . 5 |- ((D e. Met /\ (ch /\ F (_ (CC X. dom dom D))) -> (ph <-> ps))
47	46	anassrs 443	. . . 4 |- (((D e. Met /\ ch) /\ F (_ (CC X. dom dom D)) -> (ph <-> ps))
48	47	ex 373	. . 3 |- ((D e. Met /\ ch) -> (F (_ (CC X. dom dom D) -> (ph <-> ps)))
49	17, 48	syld 27	. 2 |- ((D e. Met /\ ch) -> ((ph \/ ps) -> (ph <-> ps)))
50		oibabs 656	. 2 |- (((ph \/ ps) -> (ph <-> ps)) <-> (ph <-> ps))
51	49, 50	sylib 198	1 |- ((D e. Met /\ ch) -> (ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588   i^i cin 2049   (_ wss 2050  (/)c0 2283   X. cxp 3174  dom cdm 3176  ran crn 3177   |` cres 3178   Fn wfn 3183  -->wf 3184  CCcc 5244  1c1 5247  NNcn 5308  Metcme 7786

This theorem is referenced by:  lmss 7950  causs 7952

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-enr 5178  df-nr 5179  df-0r 5183  df-1r 5184  df-c 5252  df-1 5254  df-r 5256  df-met 7790

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