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Theorem seq3f1oleml 10283
 Description: Lemma for seq3f1o 10284. This is more or less the result, but stated in terms of 𝐹 and 𝐺 without 𝐻. 𝐿 and 𝐻 may differ in terms of what happens to terms after 𝑁. The terms after 𝑁 don't matter for the value at 𝑁 but we need some definition given the way our theorems concerning seq work. (Contributed by Jim Kingdon, 17-Aug-2022.)
Hypotheses
Ref Expression
iseqf1o.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqf1o.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
iseqf1o.3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
iseqf1o.4 (𝜑𝑁 ∈ (ℤ𝑀))
iseqf1o.6 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1o.7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
iseqf1o.l 𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))
Assertion
Ref Expression
seq3f1oleml (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Distinct variable groups:   𝑥, + ,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐿,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥,𝑁,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem seq3f1oleml
Dummy variables 𝑓 𝑘 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqf1o.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2 iseqf1o.2 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3 iseqf1o.3 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
4 iseqf1o.4 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
5 iseqf1o.6 . . 3 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
6 iseqf1o.7 . . 3 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
7 iseqf1o.l . . 3 𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))
8 breq1 3932 . . . . 5 (𝑎 = 𝑥 → (𝑎𝑁𝑥𝑁))
9 2fveq3 5426 . . . . 5 (𝑎 = 𝑥 → (𝐺‘(𝑓𝑎)) = (𝐺‘(𝑓𝑥)))
108, 9ifbieq1d 3494 . . . 4 (𝑎 = 𝑥 → if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)) = if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
1110cbvmptv 4024 . . 3 (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))) = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
121, 2, 3, 4, 5, 6, 7, 11seq3f1olemp 10282 . 2 (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
13 fveq2 5421 . . . . . 6 (𝑏 = 𝑥 → (𝑓𝑏) = (𝑓𝑥))
14 id 19 . . . . . 6 (𝑏 = 𝑥𝑏 = 𝑥)
1513, 14eqeq12d 2154 . . . . 5 (𝑏 = 𝑥 → ((𝑓𝑏) = 𝑏 ↔ (𝑓𝑥) = 𝑥))
1615cbvralv 2654 . . . 4 (∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ↔ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥)
17163anbi2i 1173 . . 3 ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
18 simpr3 989 . . . 4 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
194adantr 274 . . . . 5 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑁 ∈ (ℤ𝑀))
20 elfzuz 9809 . . . . . . . 8 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ𝑀))
2120adantl 275 . . . . . . 7 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (ℤ𝑀))
22 elfzle2 9815 . . . . . . . . . 10 (𝑘 ∈ (𝑀...𝑁) → 𝑘𝑁)
2322adantl 275 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘𝑁)
2423iftrued 3481 . . . . . . . 8 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)) = (𝐺‘(𝑓𝑘)))
25 fveq2 5421 . . . . . . . . . . . 12 (𝑏 = 𝑘 → (𝑓𝑏) = (𝑓𝑘))
26 id 19 . . . . . . . . . . . 12 (𝑏 = 𝑘𝑏 = 𝑘)
2725, 26eqeq12d 2154 . . . . . . . . . . 11 (𝑏 = 𝑘 → ((𝑓𝑏) = 𝑏 ↔ (𝑓𝑘) = 𝑘))
28 simplr2 1024 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏)
29 simpr 109 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
3027, 28, 29rspcdva 2794 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑓𝑘) = 𝑘)
3130fveq2d 5425 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝑓𝑘)) = (𝐺𝑘))
32 fveq2 5421 . . . . . . . . . . 11 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
3332eleq1d 2208 . . . . . . . . . 10 (𝑥 = 𝑘 → ((𝐺𝑥) ∈ 𝑆 ↔ (𝐺𝑘) ∈ 𝑆))
346ralrimiva 2505 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
3534ad2antrr 479 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
3633, 35, 21rspcdva 2794 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ 𝑆)
3731, 36eqeltrd 2216 . . . . . . . 8 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝑓𝑘)) ∈ 𝑆)
3824, 37eqeltrd 2216 . . . . . . 7 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)) ∈ 𝑆)
39 breq1 3932 . . . . . . . . 9 (𝑎 = 𝑘 → (𝑎𝑁𝑘𝑁))
40 2fveq3 5426 . . . . . . . . 9 (𝑎 = 𝑘 → (𝐺‘(𝑓𝑎)) = (𝐺‘(𝑓𝑘)))
4139, 40ifbieq1d 3494 . . . . . . . 8 (𝑎 = 𝑘 → if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)) = if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)))
42 eqid 2139 . . . . . . . 8 (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))) = (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))
4341, 42fvmptg 5497 . . . . . . 7 ((𝑘 ∈ (ℤ𝑀) ∧ if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)) ∈ 𝑆) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑘) = if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)))
4421, 38, 43syl2anc 408 . . . . . 6 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑘) = if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)))
4544, 24, 313eqtrd 2176 . . . . 5 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑘) = (𝐺𝑘))
46 simpr 109 . . . . . . 7 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → 𝑥 ∈ (ℤ𝑀))
47 fveq2 5421 . . . . . . . . . 10 (𝑎 = (𝑓𝑥) → (𝐺𝑎) = (𝐺‘(𝑓𝑥)))
4847eleq1d 2208 . . . . . . . . 9 (𝑎 = (𝑓𝑥) → ((𝐺𝑎) ∈ 𝑆 ↔ (𝐺‘(𝑓𝑥)) ∈ 𝑆))
4934ad3antrrr 483 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
50 fveq2 5421 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝐺𝑎) = (𝐺𝑥))
5150eleq1d 2208 . . . . . . . . . . 11 (𝑎 = 𝑥 → ((𝐺𝑎) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
5251cbvralv 2654 . . . . . . . . . 10 (∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆 ↔ ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
5349, 52sylibr 133 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → ∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆)
54 simpr1 987 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
5554ad2antrr 479 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
56 f1of 5367 . . . . . . . . . . . 12 (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝑓:(𝑀...𝑁)⟶(𝑀...𝑁))
5755, 56syl 14 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑓:(𝑀...𝑁)⟶(𝑀...𝑁))
58 simpr 109 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑥𝑁)
5946adantr 274 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑥 ∈ (ℤ𝑀))
60 eluzelz 9342 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
614, 60syl 14 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℤ)
6261ad3antrrr 483 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑁 ∈ ℤ)
63 elfz5 9805 . . . . . . . . . . . . 13 ((𝑥 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥𝑁))
6459, 62, 63syl2anc 408 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥𝑁))
6558, 64mpbird 166 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑥 ∈ (𝑀...𝑁))
6657, 65ffvelrnd 5556 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → (𝑓𝑥) ∈ (𝑀...𝑁))
67 elfzuz 9809 . . . . . . . . . 10 ((𝑓𝑥) ∈ (𝑀...𝑁) → (𝑓𝑥) ∈ (ℤ𝑀))
6866, 67syl 14 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → (𝑓𝑥) ∈ (ℤ𝑀))
6948, 53, 68rspcdva 2794 . . . . . . . 8 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → (𝐺‘(𝑓𝑥)) ∈ 𝑆)
70 fveq2 5421 . . . . . . . . . 10 (𝑎 = 𝑀 → (𝐺𝑎) = (𝐺𝑀))
7170eleq1d 2208 . . . . . . . . 9 (𝑎 = 𝑀 → ((𝐺𝑎) ∈ 𝑆 ↔ (𝐺𝑀) ∈ 𝑆))
7234, 52sylibr 133 . . . . . . . . . 10 (𝜑 → ∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆)
7372ad3antrrr 483 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ ¬ 𝑥𝑁) → ∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆)
74 eluzel2 9338 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
754, 74syl 14 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
7675ad3antrrr 483 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ ¬ 𝑥𝑁) → 𝑀 ∈ ℤ)
77 uzid 9347 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
7876, 77syl 14 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ ¬ 𝑥𝑁) → 𝑀 ∈ (ℤ𝑀))
7971, 73, 78rspcdva 2794 . . . . . . . 8 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ ¬ 𝑥𝑁) → (𝐺𝑀) ∈ 𝑆)
80 eluzelz 9342 . . . . . . . . . 10 (𝑥 ∈ (ℤ𝑀) → 𝑥 ∈ ℤ)
8180adantl 275 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → 𝑥 ∈ ℤ)
8261ad2antrr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → 𝑁 ∈ ℤ)
83 zdcle 9134 . . . . . . . . 9 ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥𝑁)
8481, 82, 83syl2anc 408 . . . . . . . 8 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → DECID 𝑥𝑁)
8569, 79, 84ifcldadc 3501 . . . . . . 7 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)) ∈ 𝑆)
8610, 42fvmptg 5497 . . . . . . 7 ((𝑥 ∈ (ℤ𝑀) ∧ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)) ∈ 𝑆) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑥) = if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
8746, 85, 86syl2anc 408 . . . . . 6 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑥) = if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
8887, 85eqeltrd 2216 . . . . 5 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑥) ∈ 𝑆)
896adantlr 468 . . . . 5 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
901adantlr 468 . . . . 5 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
9119, 45, 88, 89, 90seq3fveq 10251 . . . 4 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
9218, 91eqtr3d 2174 . . 3 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
9317, 92sylan2br 286 . 2 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
9412, 93exlimddv 1870 1 (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 819   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ifcif 3474   class class class wbr 3929   ↦ cmpt 3989  ⟶wf 5119  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774   ≤ cle 7808  ℤcz 9061  ℤ≥cuz 9333  ...cfz 9797  seqcseq 10225 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-1o 6313  df-er 6429  df-en 6635  df-fin 6637  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798  df-fzo 9927  df-seqfrec 10226 This theorem is referenced by:  seq3f1o  10284
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