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Theorem rdgeqoa 33189
Description: If a recursive function with an initial value 𝐴 at step 𝑁 is equal to itself with an initial value 𝐵 at step 𝑀, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.)
Assertion
Ref Expression
rdgeqoa ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋))))

Proof of Theorem rdgeqoa
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp3 1061 . 2 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → 𝑋 ∈ ω)
2 eleq1 2687 . . . . 5 (𝑥 = 𝑋 → (𝑥 ∈ ω ↔ 𝑋 ∈ ω))
323anbi3d 1403 . . . 4 (𝑥 = 𝑋 → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω)))
4 oveq2 6643 . . . . . . 7 (𝑥 = 𝑋 → (𝑁 +𝑜 𝑥) = (𝑁 +𝑜 𝑋))
54fveq2d 6182 . . . . . 6 (𝑥 = 𝑋 → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)))
6 oveq2 6643 . . . . . . 7 (𝑥 = 𝑋 → (𝑀 +𝑜 𝑥) = (𝑀 +𝑜 𝑋))
76fveq2d 6182 . . . . . 6 (𝑥 = 𝑋 → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋)))
85, 7eqeq12d 2635 . . . . 5 (𝑥 = 𝑋 → ((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋))))
98imbi2d 330 . . . 4 (𝑥 = 𝑋 → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋)))))
103, 9imbi12d 334 . . 3 (𝑥 = 𝑋 → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋))))))
11 peano1 7070 . . . . 5 ∅ ∈ ω
12 oa0 7581 . . . . . . . . . . . 12 (𝑁 ∈ On → (𝑁 +𝑜 ∅) = 𝑁)
1312fveq2d 6182 . . . . . . . . . . 11 (𝑁 ∈ On → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐴)‘𝑁))
1413eqcomd 2626 . . . . . . . . . 10 (𝑁 ∈ On → (rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)))
15 oa0 7581 . . . . . . . . . . . 12 (𝑀 ∈ On → (𝑀 +𝑜 ∅) = 𝑀)
1615fveq2d 6182 . . . . . . . . . . 11 (𝑀 ∈ On → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘𝑀))
1716eqcomd 2626 . . . . . . . . . 10 (𝑀 ∈ On → (rec(𝐹, 𝐵)‘𝑀) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅)))
1814, 17eqeqan12d 2636 . . . . . . . . 9 ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅))))
1918biimpd 219 . . . . . . . 8 ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅))))
20 eleq1 2687 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 ∈ ω ↔ ∅ ∈ ω))
21203anbi3d 1403 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω)))
2211biantru 526 . . . . . . . . . . . 12 (𝑀 ∈ On ↔ (𝑀 ∈ On ∧ ∅ ∈ ω))
2322anbi2i 729 . . . . . . . . . . 11 ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
24 3anass 1040 . . . . . . . . . . 11 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
2523, 24bitr4i 267 . . . . . . . . . 10 ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω))
2621, 25syl6bbr 278 . . . . . . . . 9 (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On)))
27 oveq2 6643 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑁 +𝑜 𝑥) = (𝑁 +𝑜 ∅))
2827fveq2d 6182 . . . . . . . . . . 11 (𝑥 = ∅ → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)))
29 oveq2 6643 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑀 +𝑜 𝑥) = (𝑀 +𝑜 ∅))
3029fveq2d 6182 . . . . . . . . . . 11 (𝑥 = ∅ → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅)))
3128, 30eqeq12d 2635 . . . . . . . . . 10 (𝑥 = ∅ → ((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅))))
3231imbi2d 330 . . . . . . . . 9 (𝑥 = ∅ → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅)))))
3326, 32imbi12d 334 . . . . . . . 8 (𝑥 = ∅ → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅))))))
3419, 33mpbiri 248 . . . . . . 7 (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
3534ax-gen 1720 . . . . . 6 𝑥(𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
36 sbc6g 3455 . . . . . 6 (∅ ∈ ω → ([∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ∀𝑥(𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))))
3735, 36mpbiri 248 . . . . 5 (∅ ∈ ω → [∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
3811, 37ax-mp 5 . . . 4 [∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
39 peano2b 7066 . . . . 5 (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
40393anbi3i 1253 . . . . . . . 8 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω))
4140imbi1i 339 . . . . . . 7 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
42 nnon 7056 . . . . . . . . . . . . 13 (𝑥 ∈ ω → 𝑥 ∈ On)
43 oacl 7600 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ On ∧ 𝑥 ∈ On) → (𝑁 +𝑜 𝑥) ∈ On)
44 oacl 7600 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ On ∧ 𝑥 ∈ On) → (𝑀 +𝑜 𝑥) ∈ On)
4543, 44anim12i 589 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑀 ∈ On ∧ 𝑥 ∈ On)) → ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On))
46453impdir 1380 . . . . . . . . . . . . . . 15 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On) → ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On))
47 rdgsuc 7505 . . . . . . . . . . . . . . . . . 18 ((𝑁 +𝑜 𝑥) ∈ On → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥))))
48 fveq2 6178 . . . . . . . . . . . . . . . . . 18 ((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) → (𝐹‘(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥))) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
4947, 48sylan9eqr 2676 . . . . . . . . . . . . . . . . 17 (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ (𝑁 +𝑜 𝑥) ∈ On) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
5049adantrr 752 . . . . . . . . . . . . . . . 16 (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
51 rdgsuc 7505 . . . . . . . . . . . . . . . . 17 ((𝑀 +𝑜 𝑥) ∈ On → (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
5251ad2antll 764 . . . . . . . . . . . . . . . 16 (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On)) → (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
5350, 52eqtr4d 2657 . . . . . . . . . . . . . . 15 (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
5446, 53sylan2 491 . . . . . . . . . . . . . 14 (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
5554ancoms 469 . . . . . . . . . . . . 13 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
5642, 55syl3anl3 1374 . . . . . . . . . . . 12 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
57 onasuc 7593 . . . . . . . . . . . . . . 15 ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (𝑁 +𝑜 suc 𝑥) = suc (𝑁 +𝑜 𝑥))
5857fveq2d 6182 . . . . . . . . . . . . . 14 ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)))
59583adant2 1078 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)))
6059adantr 481 . . . . . . . . . . . 12 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)))
61 onasuc 7593 . . . . . . . . . . . . . . 15 ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (𝑀 +𝑜 suc 𝑥) = suc (𝑀 +𝑜 𝑥))
6261fveq2d 6182 . . . . . . . . . . . . . 14 ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
63623adant1 1077 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
6463adantr 481 . . . . . . . . . . . 12 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
6556, 60, 643eqtr4d 2664 . . . . . . . . . . 11 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))
6665ex 450 . . . . . . . . . 10 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))
6766imim2d 57 . . . . . . . . 9 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
6840, 67sylbir 225 . . . . . . . 8 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
6968a2i 14 . . . . . . 7 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
7041, 69sylbi 207 . . . . . 6 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
71 sbcimg 3471 . . . . . . 7 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → [suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))))
72 sbc3an 3488 . . . . . . . . 9 ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ ([suc 𝑥 / 𝑥]𝑁 ∈ On ∧ [suc 𝑥 / 𝑥]𝑀 ∈ On ∧ [suc 𝑥 / 𝑥]𝑥 ∈ ω))
73 sbcg 3497 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑁 ∈ On ↔ 𝑁 ∈ On))
74 sbcg 3497 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑀 ∈ On ↔ 𝑀 ∈ On))
75 sbcel1v 3489 . . . . . . . . . . 11 ([suc 𝑥 / 𝑥]𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
7675a1i 11 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑥 ∈ ω ↔ suc 𝑥 ∈ ω))
7773, 74, 763anbi123d 1397 . . . . . . . . 9 (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥]𝑁 ∈ On ∧ [suc 𝑥 / 𝑥]𝑀 ∈ On ∧ [suc 𝑥 / 𝑥]𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω)))
7872, 77syl5bb 272 . . . . . . . 8 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω)))
79 sbcimg 3471 . . . . . . . . 9 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → [suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
80 sbcg 3497 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) ↔ (rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀)))
81 sbceqg 3975 . . . . . . . . . . 11 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ suc 𝑥 / 𝑥(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = suc 𝑥 / 𝑥(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
82 csbfv12 6218 . . . . . . . . . . . . 13 suc 𝑥 / 𝑥(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (suc 𝑥 / 𝑥rec(𝐹, 𝐴)‘suc 𝑥 / 𝑥(𝑁 +𝑜 𝑥))
83 csbconstg 3539 . . . . . . . . . . . . . 14 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥rec(𝐹, 𝐴) = rec(𝐹, 𝐴))
84 csbov123 6672 . . . . . . . . . . . . . . 15 suc 𝑥 / 𝑥(𝑁 +𝑜 𝑥) = (suc 𝑥 / 𝑥𝑁suc 𝑥 / 𝑥 +𝑜 suc 𝑥 / 𝑥𝑥)
85 csbconstg 3539 . . . . . . . . . . . . . . . 16 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥 +𝑜 = +𝑜 )
86 csbconstg 3539 . . . . . . . . . . . . . . . 16 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥𝑁 = 𝑁)
87 csbvarg 3994 . . . . . . . . . . . . . . . 16 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥𝑥 = suc 𝑥)
8885, 86, 87oveq123d 6656 . . . . . . . . . . . . . . 15 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥𝑁suc 𝑥 / 𝑥 +𝑜 suc 𝑥 / 𝑥𝑥) = (𝑁 +𝑜 suc 𝑥))
8984, 88syl5eq 2666 . . . . . . . . . . . . . 14 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥(𝑁 +𝑜 𝑥) = (𝑁 +𝑜 suc 𝑥))
9083, 89fveq12d 6184 . . . . . . . . . . . . 13 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥rec(𝐹, 𝐴)‘suc 𝑥 / 𝑥(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)))
9182, 90syl5eq 2666 . . . . . . . . . . . 12 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)))
92 csbfv12 6218 . . . . . . . . . . . . 13 suc 𝑥 / 𝑥(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) = (suc 𝑥 / 𝑥rec(𝐹, 𝐵)‘suc 𝑥 / 𝑥(𝑀 +𝑜 𝑥))
93 csbconstg 3539 . . . . . . . . . . . . . 14 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥rec(𝐹, 𝐵) = rec(𝐹, 𝐵))
94 csbov123 6672 . . . . . . . . . . . . . . 15 suc 𝑥 / 𝑥(𝑀 +𝑜 𝑥) = (suc 𝑥 / 𝑥𝑀suc 𝑥 / 𝑥 +𝑜 suc 𝑥 / 𝑥𝑥)
95 csbconstg 3539 . . . . . . . . . . . . . . . 16 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥𝑀 = 𝑀)
9685, 95, 87oveq123d 6656 . . . . . . . . . . . . . . 15 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥𝑀suc 𝑥 / 𝑥 +𝑜 suc 𝑥 / 𝑥𝑥) = (𝑀 +𝑜 suc 𝑥))
9794, 96syl5eq 2666 . . . . . . . . . . . . . 14 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥(𝑀 +𝑜 𝑥) = (𝑀 +𝑜 suc 𝑥))
9893, 97fveq12d 6184 . . . . . . . . . . . . 13 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥rec(𝐹, 𝐵)‘suc 𝑥 / 𝑥(𝑀 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))
9992, 98syl5eq 2666 . . . . . . . . . . . 12 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))
10091, 99eqeq12d 2635 . . . . . . . . . . 11 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = suc 𝑥 / 𝑥(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))
10181, 100bitrd 268 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))
10280, 101imbi12d 334 . . . . . . . . 9 (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → [suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
10379, 102bitrd 268 . . . . . . . 8 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
10478, 103imbi12d 334 . . . . . . 7 (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → [suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))))
10571, 104bitrd 268 . . . . . 6 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))))
10670, 105syl5ibr 236 . . . . 5 (suc 𝑥 ∈ ω → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) → [suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))))
10739, 106sylbi 207 . . . 4 (𝑥 ∈ ω → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) → [suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))))
10838, 107findes 7081 . . 3 (𝑥 ∈ ω → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
10910, 108vtoclga 3267 . 2 (𝑋 ∈ ω → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋)))))
1101, 109mpcom 38 1 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1479   = wceq 1481  wcel 1988  [wsbc 3429  csb 3526  c0 3907  Oncon0 5711  suc csuc 5713  cfv 5876  (class class class)co 6635  ωcom 7050  reccrdg 7490   +𝑜 coa 7542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549
This theorem is referenced by:  finxpreclem4  33202
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