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Theorem gsumval3lem1 19689
Description: Lemma 1 for gsumval3 19691. (Contributed by AV, 31-May-2019.)
Hypotheses
Ref Expression
gsumval3.b 𝐡 = (Baseβ€˜πΊ)
gsumval3.0 0 = (0gβ€˜πΊ)
gsumval3.p + = (+gβ€˜πΊ)
gsumval3.z 𝑍 = (Cntzβ€˜πΊ)
gsumval3.g (πœ‘ β†’ 𝐺 ∈ Mnd)
gsumval3.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
gsumval3.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
gsumval3.c (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
gsumval3.m (πœ‘ β†’ 𝑀 ∈ β„•)
gsumval3.h (πœ‘ β†’ 𝐻:(1...𝑀)–1-1→𝐴)
gsumval3.n (πœ‘ β†’ (𝐹 supp 0 ) βŠ† ran 𝐻)
gsumval3.w π‘Š = ((𝐹 ∘ 𝐻) supp 0 )
Assertion
Ref Expression
gsumval3lem1 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))
Distinct variable groups:   + ,𝑓   𝐴,𝑓   πœ‘,𝑓   𝑓,𝐺   𝑓,𝑀   𝐡,𝑓   𝑓,𝐹   𝑓,𝐻   𝑓,π‘Š
Allowed substitution hints:   𝑉(𝑓)   0 (𝑓)   𝑍(𝑓)

Proof of Theorem gsumval3lem1
StepHypRef Expression
1 gsumval3.h . . . . . . 7 (πœ‘ β†’ 𝐻:(1...𝑀)–1-1→𝐴)
21ad2antrr 725 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝐻:(1...𝑀)–1-1→𝐴)
3 gsumval3.w . . . . . . . . 9 π‘Š = ((𝐹 ∘ 𝐻) supp 0 )
4 suppssdm 8113 . . . . . . . . 9 ((𝐹 ∘ 𝐻) supp 0 ) βŠ† dom (𝐹 ∘ 𝐻)
53, 4eqsstri 3983 . . . . . . . 8 π‘Š βŠ† dom (𝐹 ∘ 𝐻)
6 gsumval3.f . . . . . . . . 9 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
7 f1f 6743 . . . . . . . . . 10 (𝐻:(1...𝑀)–1-1→𝐴 β†’ 𝐻:(1...𝑀)⟢𝐴)
81, 7syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐻:(1...𝑀)⟢𝐴)
9 fco 6697 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐻:(1...𝑀)⟢𝐴) β†’ (𝐹 ∘ 𝐻):(1...𝑀)⟢𝐡)
106, 8, 9syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (𝐹 ∘ 𝐻):(1...𝑀)⟢𝐡)
115, 10fssdm 6693 . . . . . . 7 (πœ‘ β†’ π‘Š βŠ† (1...𝑀))
1211ad2antrr 725 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ π‘Š βŠ† (1...𝑀))
13 f1ores 6803 . . . . . 6 ((𝐻:(1...𝑀)–1-1→𝐴 ∧ π‘Š βŠ† (1...𝑀)) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š))
142, 12, 13syl2anc 585 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š))
153imaeq2i 6016 . . . . . . 7 (𝐻 β€œ π‘Š) = (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 ))
16 gsumval3.a . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ 𝑉)
176, 16fexd 7182 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 ∈ V)
18 ovex 7395 . . . . . . . . . . . 12 (1...𝑀) ∈ V
19 fex 7181 . . . . . . . . . . . 12 ((𝐻:(1...𝑀)⟢𝐴 ∧ (1...𝑀) ∈ V) β†’ 𝐻 ∈ V)
207, 18, 19sylancl 587 . . . . . . . . . . 11 (𝐻:(1...𝑀)–1-1→𝐴 β†’ 𝐻 ∈ V)
211, 20syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐻 ∈ V)
22 f1fun 6745 . . . . . . . . . . . 12 (𝐻:(1...𝑀)–1-1→𝐴 β†’ Fun 𝐻)
231, 22syl 17 . . . . . . . . . . 11 (πœ‘ β†’ Fun 𝐻)
24 gsumval3.n . . . . . . . . . . 11 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† ran 𝐻)
2523, 24jca 513 . . . . . . . . . 10 (πœ‘ β†’ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻))
2617, 21, 25jca31 516 . . . . . . . . 9 (πœ‘ β†’ ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)))
2726ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)))
28 imacosupp 8145 . . . . . . . . 9 ((𝐹 ∈ V ∧ 𝐻 ∈ V) β†’ ((Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 )))
2928imp 408 . . . . . . . 8 (((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
3027, 29syl 17 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
3115, 30eqtrid 2789 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ π‘Š) = (𝐹 supp 0 ))
3231f1oeq3d 6786 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š) ↔ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 )))
3314, 32mpbid 231 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 ))
34 isof1o 7273 . . . . 5 (𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š) β†’ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
3534ad2antll 728 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š)
36 f1oco 6812 . . . 4 (((𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 ) ∧ 𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š) β†’ ((𝐻 β†Ύ π‘Š) ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ))
3733, 35, 36syl2anc 585 . . 3 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 β†Ύ π‘Š) ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ))
38 f1of 6789 . . . . 5 (𝑓:(1...(β™―β€˜π‘Š))–1-1-ontoβ†’π‘Š β†’ 𝑓:(1...(β™―β€˜π‘Š))βŸΆπ‘Š)
39 frn 6680 . . . . 5 (𝑓:(1...(β™―β€˜π‘Š))βŸΆπ‘Š β†’ ran 𝑓 βŠ† π‘Š)
4035, 38, 393syl 18 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ran 𝑓 βŠ† π‘Š)
41 cores 6206 . . . 4 (ran 𝑓 βŠ† π‘Š β†’ ((𝐻 β†Ύ π‘Š) ∘ 𝑓) = (𝐻 ∘ 𝑓))
42 f1oeq1 6777 . . . 4 (((𝐻 β†Ύ π‘Š) ∘ 𝑓) = (𝐻 ∘ 𝑓) β†’ (((𝐻 β†Ύ π‘Š) ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 )))
4340, 41, 423syl 18 . . 3 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (((𝐻 β†Ύ π‘Š) ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 )))
4437, 43mpbid 231 . 2 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ))
45 fzfi 13884 . . . . . . 7 (1...𝑀) ∈ Fin
46 ssfi 9124 . . . . . . 7 (((1...𝑀) ∈ Fin ∧ π‘Š βŠ† (1...𝑀)) β†’ π‘Š ∈ Fin)
4745, 11, 46sylancr 588 . . . . . 6 (πœ‘ β†’ π‘Š ∈ Fin)
4847ad2antrr 725 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ π‘Š ∈ Fin)
493a1i 11 . . . . . . . . 9 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ π‘Š = ((𝐹 ∘ 𝐻) supp 0 ))
5049imaeq2d 6018 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ π‘Š) = (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )))
5145a1i 11 . . . . . . . . . . . 12 (πœ‘ β†’ (1...𝑀) ∈ Fin)
528, 51fexd 7182 . . . . . . . . . . 11 (πœ‘ β†’ 𝐻 ∈ V)
5317, 52, 25jca31 516 . . . . . . . . . 10 (πœ‘ β†’ ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)))
5453ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐹 ∈ V ∧ 𝐻 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 supp 0 ) βŠ† ran 𝐻)))
5554, 29syl 17 . . . . . . . 8 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ ((𝐹 ∘ 𝐻) supp 0 )) = (𝐹 supp 0 ))
5650, 55eqtrd 2777 . . . . . . 7 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β€œ π‘Š) = (𝐹 supp 0 ))
5756f1oeq3d 6786 . . . . . 6 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐻 β€œ π‘Š) ↔ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 )))
5814, 57mpbid 231 . . . . 5 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 β†Ύ π‘Š):π‘Šβ€“1-1-ontoβ†’(𝐹 supp 0 ))
5948, 58hasheqf1od 14260 . . . 4 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (β™―β€˜π‘Š) = (β™―β€˜(𝐹 supp 0 )))
6059oveq2d 7378 . . 3 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (1...(β™―β€˜π‘Š)) = (1...(β™―β€˜(𝐹 supp 0 ))))
6160f1oeq2d 6785 . 2 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ ((𝐻 ∘ 𝑓):(1...(β™―β€˜π‘Š))–1-1-ontoβ†’(𝐹 supp 0 ) ↔ (𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 )))
6244, 61mpbid 231 1 (((πœ‘ ∧ π‘Š β‰  βˆ…) ∧ (Β¬ 𝐴 ∈ ran ... ∧ 𝑓 Isom < , < ((1...(β™―β€˜π‘Š)), π‘Š))) β†’ (𝐻 ∘ 𝑓):(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  dom cdm 5638  ran crn 5639   β†Ύ cres 5640   β€œ cima 5641   ∘ ccom 5642  Fun wfun 6495  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501   Isom wiso 6502  (class class class)co 7362   supp csupp 8097  Fincfn 8890  1c1 11059   < clt 11196  β„•cn 12160  ...cfz 13431  β™―chash 14237  Basecbs 17090  +gcplusg 17140  0gc0g 17328  Mndcmnd 18563  Cntzccntz 19102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-hash 14238
This theorem is referenced by:  gsumval3lem2  19690
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